The Converse of Abel’s Theorem

by

Veniamine Kissounko

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto

Copyright c 2009 by Veniamine Kissounko Abstract

The Converse of Abel’s Theorem

Veniamine Kissounko

Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

2009

In my thesis I investigate an algebraization problem. The simplest, but already non- trivial, problem in this direction is to find necessary and sufficient conditions for three graphs of smooth functions on a given interval to belong to an algebraic curve of degree three.

The analogous problems were raised by Lie and Darboux in connection with the classification of surfaces of double translation; by Poincare and Mumford in connection with the Schottky problem; by Griffiths and Henkin in connection with a converse of

Abel’s theorem; by Bol and Akivis in the connection with the algebraization problem in the theory of webs. Interestingly, the complex-analytic technique developed by Griftiths and Henkin for the holomorphic case failed to work in the real smooth setting.

In the thesis I develop a technique of, what I call, complex moments. Together with a simple differentiation rule it provides a unified approach to all the algebraization problems considered so far (both complex-analytic and real smooth). As a result I prove two variants (’polynomial’ and ’rational’) of a converse of Abel’s theorem which significantly generalize results of Griffiths and Henkin. Already the ’polynomial’ case is nontrivial leading to a new relation between the algebraization problem in the theory of webs and the converse of Abel’s theorem.

But, perhaps, the most interesting is the rational case as a new phenomenon occurs: there are forms with logarithmic singularities on special algebraic varieties that satisfy the

ii converse of Abel’s theorem. In the thesis I give a complete description of such varieties and forms.

iii Dedication

To my Father

iv Acknowledgements

I would like to express my gratitude to all those who gave me the possibility to complete this thesis.

I want to thank the Department of Mathematics at the University of Toronto for hav- ing me as a graduate student and, especially, our graduate coordinator Ida for constant help and attention.

I am deeply indebted to my supervisor Professor Khovanskii for stating the problem, numerous stimulating discussions, encouragement and support throughout my graduate student life.

It is a pleasure to acknowledge V.Timorin, the former student of Khovanskii, for a few but very interesting conversations.

I would like to give my special thanks to my wife Xiaohan whose patient love enabled me to complete this work.

v Contents

1 Introduction and organization of material 1

1.1 Introduction ...... 1

1.2 Organization of the material ...... 2

2 Rational functions, moment-type problems, applications 8

2.1 Rational functions and a moment-type problem ...... 8

2.2 Applications ...... 15

2.2.1 Structure of finite mappings ...... 15

2.2.2 Moment-type problem and Galois theory ...... 23

2.2.3 A GAGA-type theorem ...... 26

3 Abel’s theorem 32

3.1 A version of Abel’s theorem for abstract and plane curves ...... 32

3.2 Application of Abel’s theorem: plane geometry and Euler-Jacobi formula 44

3.3 Abel’s theorem: continuation ...... 48

3.4 Multidimensional Abel’s theorem and generalized holomorphic forms . . 53

4 Differentiation rule 62

4.1 One-dimensional case ...... 62

4.2 Multidimensional case ...... 64

5 Main theorems: Converse of Abel’s theorem - Polynomial case 69

vi 6 Proofs of Main theorems 73

6.1 Sufficiency: Theorem 5.1 and Theorem 5.2 ...... 73

6.2 Necessity: Theorem 5.1 ...... 75

6.3 Necessity: Theorem 5.2 ...... 81

7 Applications of Main theorems and our methods 85

7.1 Preparatory Theorems for Chapter 9 ...... 85

7.2 Generalized holomorphic forms on complete intersections in Cn and CP n 87 7.3 Converse of Abel’s theorem - Rational case ...... 89

8 Cotes’ theorem an its converse 104

8.1 Cotes’ theorem ...... 104

8.2 The converse of Cotes’ theorem ...... 106

9 Hypersurfaces of double translation 109

9.1 Surfaces of double translation ...... 109

9.2 Multidimensional case ...... 114

9.3 Torelli theorem ...... 120

10 Appendix 125

Bibliography 132

vii Chapter 1

Introduction and organization of material

1.1 Introduction

Abel proved in [1] that, under a holomorphic mapping of a complex algebraic curve onto the Riemann sphere, the trace of a meromorphic form is meromorphic, while the trace of a holomorphic form is identically zero. Newton’s student Cotes noted the following property of a plane algebraic curve. Let the number of intersection points of a curve with each straight line from a family of parallel straight lines be d (counting multiplicities), which is the degree of the curve. On each line of the family, we mark the center of masses of the resulting d intersection points. Then all the marked centers of masses lie on a single line. The Cotes theorem is automatically extended to algebraic hypersurfaces.

In 1892, Lie [11] classified all the surfaces of double translation in a three-dimensional space. This classification is based on the Lie criterion for the algebraizability of a plane curve. The Lie criterion is the converse of Cotes’ theorem. Darboux [6] simplified the proof of the Lie theorem. He found another algebraizability criterion for a plane curve that is the converse of Abel’s theorem for holomorphic forms. Various converses were

1 Chapter 1. Introduction and organization of material 2 found by Bol [3], Griffiths [8], and Henkin [9]. Wood [19] devised a very simple proof for the converse of Cotes’ theorem for hypersurfaces, which was extended by Akivis [2] to subvarieties of any codimension. (Both Wood and Akivis dealt with the differential- geometric form of Cotes’ theorem as derived by Gauss’ student Reiss [14], but the original version of this theorem is more convenient for our purposes.)

The converses of the Abel and Cotes theorems were treated as entirely different un- related assertions. We prove a converse of Abel’s theorem for holomorphic forms that straightforwardly implies the converse of Cotes’ theorem. Our proof is nearly as simple as Wood’s original argument. It is based on a general assertion (see Theorem 2.8) and differential calculus (see Proposition 4.6). Moreover, our proof can be extended without changes to the complex-analytic case.

As further development of our methods, we prove a converse of Abel’s theorem for meromorphic forms. Intriguingly, our converse of Abel’s theorem for meromorphic forms unravels a new phenomenon comparing to the results of Henkin in [9]. In contrast to [9], precisely one extra case occurs – a finite union of lines passing through a single point on the plane. It turns out, that on a latter curve (more exactly on its normalization) there are forms with logarithmic singularities that satisfies the Abel theorem for meromorphic forms. Our theorem also provides the explicit description of all such forms.

In the manuscript, it is offen that the proof of an assertion involving real manifolds, smooth forms, etc can be extended without changes to a complex-analytic counterpart which we would like to use. In this case we state the complex-analytic counterpart without giving the proof.

1.2 Organization of the material

In the section 2.1 of Chapter 2, first we discuss how to reconstruct a rational function from the finite number of terms in its Taylor expansion at infinity. The function is Chapter 1. Introduction and organization of material 3 regular and equal to zero at infinity. We present formulas which go back to Jacobi’s paper on a rational interpolation [4], to thesis of Pade [13] in which he developed the theory of so-called Pade approximants, and later to the famous treatise on continues fractions of Stieltjes [16] in which he formulated and solved the moment problem on the positive semi-axis. As a by-product, we obtain the Kronecker criterion that describes all the sequences of complex numbers that occur as the coefficients in Taylor expansion at infinity of rational functions as above.

Interestingly, a certain moment problem, similar to a problem in the theory of mo- ments, that arises in our discussion on the reconstruction of rational functions. It is the explicit solution of this moment problem that first yields to Theorem 2.8 and then to

Proposition 2.18 which are crucial in our proofs of the converse of Abel’s theorem for holomorphic and meromorphic forms, respectively.

In the following three subsections we collect various applications of the results in

Section 2.1. In fact, this is not a complete list of possible applications as, for instance, the connection with the classical problems in the theory of moments is not presented.

In Subsection 2.2.1 we prove Theorem 2.8. This is a very general assertion which is valid over any field. Then we prove a certain topological version of Theorem 2.8 for branch coverings. As a corollary, we obtain a sort of Weierstrass’ preparation theorem in the category of topological spaces (see Corollary 2.15). It is worth mentioning that locally a purely dimensional analytic space admits a branch covering over a small open disk in some Cn. In Subsection 2.2.2, we exploit the fact that the moment problem and Theorem 2.8 makes sense over an arbitrary field. We start by proving a certain lemma from Galois theory and finish with Proposition 2.18 which has a flavor of differential Galois theory.

Proposition 2.18 is the key statement in the proof of the converse of Abel’s theorem for meromorphic forms.

In Subsection 2.2.3 we prove two technical proposition which we need further. The Chapter 1. Introduction and organization of material 4

first asserts that the degrees of rational functions on a complex line that belong to a family

N of rational functions Fλ holomorphically depending on λ ∈ C are bounded above by a single constant. We use the Kronecker criterion to show that. The second proposition

asserts that rational functions on Rn (resp. Cn) are the only functions on Rn (resp. Cn) which restriction to sufficiently many lines in Rn (resp. Cn) is rational. For the precise statements see Proposition 2.20 and Proposition 2.21, respectively. As a consequence we obtain certain GAGA-type theorems in Corollary 2.23 and in Corollary 2.24. The latter statement is taken from [8].

In Chapter 3.1 we discuss Abel theorem in its various forms, and present several applications. In the proof of the multidimensional Abel Theorem 3.24 we follow the approach in [8]. We then generalize Theorem 3.24 to residue forms on locally complete intersection, see Theorem 3.28. We finish the chapter by defining certain generalized holomorphic forms on purely dimensional affine and projective varieties over the field of complex numbers. The definition is inspired by the trace property of holomorphic forms; and for non-singular varieties, the generalized forms coincide with usual top degree holomorphic forms.

In Chapter 4 we develop the differential calculus on real manifolds needed in the proof of our converses of Abel’s theorem. We present two methods. The first is a direct corollary from Lemma 4.2 taken from [8], while the second method is based on a certain Liouville- type formula. We demonstrate the first and the second approaches by applying them to the one-dimensional case and the multi-dimensional case, respectively. In fact, the second method is more general as it allows to compute the derivative in Proposition 4.6 even when the form ω depends on the differentiation parameter t.

In Chapter 5 we state the main theorems, the converses of Abel’s theorem (for holo- morphic forms). Let us introduce some of the main characters in our converse of Abel’s

n k 1 theorem now. Suppose that surfaces γ1, . . . , γd ⊂ R × R are the graphs of C smooth

n 1 vector-valued functions defined on a connected domain in R . Let ω1, . . . , ωd be C Chapter 1. Introduction and organization of material 5

smooth forms of the highest degree on γ1, . . . , γd that vanish on nowhere dense subsets.

Our converses states that, under some assumptions on the trace of ω1, . . . , ωd under paral- lel projections along k-dimensional planes in Rn ×Rk that are sufficiently close to vertical one, the surfaces belong to an algebraic variety Γ of degree d and the forms are the re-

striction of a generalized holomorphic form on the complexification of Γ. It is convenient

to distinguish two cases: the case of codimension one (k = 1) and the case of the higher

codimension (k > 1). In the first case our proof immediately shows that the generalized

holomorphic form that arises is a residue form. See Theorem 5.1 and 5.2, respectively,

for the precise statements.

In Chapter 6 we present the proofs of our main theorems. Several words on the

proofs. The sufficiency in Theorem 5.1 and Theorem 5.2 are rather straightforward. By

projecting to the affine spaces of lower dimension, the algebraization of the surfaces in the

necessity of Theorem 5.2 easily follows from the codimension one case; we also conclude

that the forms ω1, . . . , ωd are the restriction of a single rational n-form Ω on the algebraic

variety Γ ⊂ Rn+k of degree d that contains the surfaces. Now, when the forms do not

vanish on γ1, . . . , γd, the necessity of Theorem 5.1 is a combination of the differential calculus and the Proposition 6.3, which is the direct consequence of Theorem 2.8. By

slightly modifing the proof, we include the case when the forms ω1, . . . , ωd do vanish on the surfaces.

Essentially, using the differential calculus we show that the forms ω1, . . . , ωd in both

theorems constitute the module over the ring of polynomials on Rn+k. The latter together

with a Theorem 10.1 from the appendix shows that the form Ω on Γ ⊂ Rn+k is, in fact, a generalized holomorphic form on the complexification of Γ.

In Chapter 7 we present several corollaries from the main theorems and our methods.

In the first section we present a corollary from Theorem 5.2 and a corollary from its

complex-analytic counterpart. In the second section we, essentially, establish the con-

verses of Corollaries 3.31, 3.32, thus proving that the generalized holomorphic forms on Chapter 1. Introduction and organization of material 6 complete intersections in Cn and CP n are the residue forms. The results in the third section of Chapter 7 are, probably, the most interesting.

We prove Theorem 7.9 which is the converse of Abel’s theorem for meromorphic forms in codimension one. The generalization to the case of codimension more than one is straightforward. As mentioned in the introduction, our theorem is very different from the similar result in [9] since we have the following exceptional case. On the finite union hyperplanes in Cn+1 passing through a single plane of codimension 2 there are top forms with logarithmic singularities that satisfy the Abel theorem. For the ease of exposition we deal with the codimension one, complex-analytic version of the theorem.

The algebraization of the surfaces in Theorem 7.9 is a combination of the complex- analytic counterpart of the differential calculus in Proposition 4.6 and Proposition 7.5, which is the corollary of Proposition 2.18.

At this point it is instructive to examine the difference between the key assertions, besides the differential calculus, in the proofs of the converse Abel’s theorem for holomor- phic and meromorphic forms. Mainly, the difference lies in the following two propositions.

Let K0 ⊂ K be fields and ρ1, . . . , ρd non-zero K-valued masses located at the pairwise distinct elements y1, . . . , yd of the field K. Suppose that the first 2d moments of the system of masses belong to K0, then y1, . . . , yd are the roots of a polynomial of degree d over the field K0. This the key assertions for the holomorphic case.

Let K0 ⊂ K ⊂ L be fields and a group G acts on K with K0 the field of invariants.

Suppose that ρ1, . . . , ρd are non-zero L-valued masses located at the pairwise distinct elements y1, . . . , yd of the field L. Suppose that the first 4d moments of the system of masses belong to K. Under a certain condition on how the group G acts on the moments

(see Proposition 2.18), the elements y1, . . . , yd are the roots of a polynomial of degree d over the field K0. This the key assertions for the meromorphic case.

We obtain the description of the forms ω1, . . . , ωd in Theorem 7.9 in the following way. First, we prove the case n = 1. Then we use appropriate plane sections and Chapter 1. Introduction and organization of material 7

Proposition 7.5 to reduce the case n > 1 to the case above.

In Chapter 8 we discuss the Cotes theorem and its converse.

In Chapter 9 we discuss hypersurfaces of double translation in Rn. A hypersurface of double translation is a smooth surface that admits two distinct translation structures (see

Chapter 9 for the precise definitions). We prove the Lie classification theorem in the three- dimensional space and its multidimensional generalization due to Wirtinger [18]. We also show that a hypersurfaces of double translation does not allow for a third translation structure. The complex-analytic counterpart of the latter statement leads to the proof of a version of the celebrated Torelli theorem for curves that we present in section 9.3.

This connection was first found by B. Saint-Donat in [15]; see also [5] for a survey on this fascinating subject. It is interesting to mention that our version has a local flavor: given a double translation structure of a certain analytic hypersurface that is defined in a neighborhood of a point in Cg, we reconstruct the non-hyperelliptic Riemann surface

S of the genus g. More precisely: the analytic hypersurface in question is the lifting of a germ of Ab(S(g−1)) ⊂ Cg\Λ to Cg, where Cg\Λ is the jacobian of the Reimann surface and Ab(S(g−1)) is the image of the (g − 1) symmetric power of the curve S under the

Abel-Jacobi mapping. It is also worth to emphasize that in our version the lattice Λ play no role. A priori, we do not require the factor space Cg\Λ to be a compact complex torus. This is why our arguments will work, for instance, in the case of non-hyperelliptic irreducible Gorenshtein curves.

In the last Chapter 10, we use the machinery of Hefer polynomials to prove Theo- rem 10.1. This theorem is vital in the necessity of Theorem 5.2 to show that the rational form Ω on the algebraic variety Γ, the restriction of which to the surfaces γ1, . . . , γd coincide ω1, . . . , ωd, is, in fact, generalized holomorphic. Interestingly, to show Theo- rem 10.1 we use a multidimensional version of the formula in Theorem 2.1(2) for the polynomial-numerator. Chapter 2

Rational functions, moment-type

problems, and applications

2.1 Rational functions and a moment-type problem

Let F be a rational function given as a quotient P/Q of two relatively prime polynomials

P and Q with complex coefficients. We assume that Q is a monic polynomial of degree

n (n ∈ N) and P is a polynomial of degree strictly less than n. In other words, P =

n−1 n n−1 bn−1z + ... + b0 and Q = z + an−1z + ... + a0, where ai, bj - are complex numbers. ∞ P sk Consider Taylor series of the function F at infinity: F = zk . In the first theorem we k=1 discuss how to reconstruct coefficients of polynomials P and Q from 2n complex numbers s1, . . . , s2n. In the second theorem we answer the converse question: given a 2n-tuple of 2n P sk −2n complex numbers is there a rational function F such that F (z) = zk + o(z ) in the k=1 neighborhood of infinity?

Let us fix some denotations:

8 Chapter 2. Rational functions, moment-type problems, applications 9

    s1 s2 . . . sn sn+1   s1 s2 . . . sn     s s . . . s s     2 3 n+1 n+2 s2 s3 . . . sn+1   . . .  ∆ =   , ∆(z) =  ......   . . . .   . . .   . . .. .          sn sn+1 . . . s2n−1 s2n    sn sn+1 . . . s2n−1   1 z . . . zn−1 zn

Theorem 2.1. 1. The determinant of the matrix ∆ is not zero.

2. Polynomial Q(z) is equal to det ∆(z) / det ∆.

3. If (bn−1, . . . , b0) and (1, an−1, . . . , a1) are vectors constructed from coefficients of polynomials P and Q respectively then       bn−1 s1 0 0 ... 0 1             bn−2 s2 s1 0 ... 0  an−1   =      .   . . . . .   .   .   ......   .              b0 sn sn−1 sn−2 . . . s1 a1

Theorem 2.2. For any 2n-tuple of complex numbers (s1, . . . , s2n) such that det ∆ 6= 0, 2n P sk −2n there exists a unique rational function F of degree n with F (z) = zk + o(z ) in a k=1 neighborhood of infinity. Polynomials P and Q associated with the function (F = P/Q)

are given by the formulas in Theorem 2.1 (2) and (3).

Remark 2.1. Theorems 2.1 and 2.2 are very general. They remain valid over a field of an 2n P sk −2n arbitrary characteristic. The identity F = P/Q = zk + o(z ) should be understood k=1 as an identity in the ring of formal power series.

For the proof of Theorem 2.1 we need a lemma.

Lemma 2.3. If two regular at infinity rational functions of degree at most n have the

same Taylor series at infinity up to the order 2n + 1, then they coincide.

P P1 Proof of Lemma 2.3. Assume that rational functions F = and F1 = satisfy the Q Q1

lemma: deg P ≤ deg Q ≤ n, deg P1 ≤ deg Q1 ≤ n. The difference F − F1 has a Chapter 2. Rational functions, moment-type problems, applications 10

zero at infinity of order at least 2n + 1. On the other hand, note that the function

P P3 PQ1−QP1 F − F1 = − = has a zero of order at most 2n, unless it is identically equal Q Q1 QQ1

to zero. Therefore F = F1.

Proofs of Theorems 2.1 and 2.2. Expanding brackets in the identity P (z) = Q(z)·  2n  P sk −2n −n −1 0 1 n−1 zk + o(z ) and collecting coefficients before z , . . . , z , z , z , . . . , z , we ob- k=1 tain a system of linear non-homogeneous equations with (an−1, . . . , a0) and (bn−1, . . . , b0) unknowns. Let us call this system S. Any solution of the system S gives rise to poly-

n−1 n n−1 nomials P = bn−1z + ... + b0 and Q = z + an−1z + ... + a0 with the quotient 2n P sk −k F = P/Q equal to zk + o(z ) in a neighborhood of infinity. Indeed, consider the k=1 „ 2n s «  2n  2n P −Q P k zk P sk P P sk k=1 rational function G = F − zk = Q − ( zk ) = Q . The numerator of k=1 k=1 −n−1 −2n the latter expression has the form c−n−1z + ... + c−2nz with ci ∈ C, and thus it has a zero of order at least n + 1 at infinity. The denominator has a pole of order n at

infinity. Therefore G has a zero at infinity of order at least 2n + 1, which implies that 2n P P sk −2n F = Q = zk + o(z ) in a neighborhood of infinity. More explicitly, the system S k=1 has the following form:   ∆ 0     x = −c, (2.1) C −E where c, x are vectors (sn+1, . . . , s2n, s1, . . . , sn) and (a0, . . . , an−1, bn−1, . . . , b0) respec- tively, and C - is a n × n matrix.

P 1. Now, given the rational function F = Q , the system S has a solution. Any other solution gives rise to a pair of polynomials P1 and Q1 which, according to Lemma 2.3, defines the same rational function: P1 = P . We recall that polynomials P and Q are Q1 Q relatively prime and the leading coefficients of polynomials Q and Q1 are equal to 1.

Thus P = P1 and Q = Q1. Conclusion: the system S has a unique solution. Therefore its determinant, which up to a sign coincide with det ∆, is not equal to zero.

Now we verify the formula in Theorem 2.1(2) for the polynomial Q. Denote the Chapter 2. Rational functions, moment-type problems, applications 11

n-th column of the matrix ∆ by vn and vector (sn+1, . . . , s2n) by c. According to the

det(v1,...,−c,...,vn) Cramer’s rule, ak = det ∆ with the vector c inserted on the k-th place. So n n−1 n n−1 det ∆Q = det ∆(z + an−1z + ... + a0) = det ∆z + det(v1, v2,..., −c)z + ... + det(−c, v2, . . . , . . . , vn). The latter expression coincide with the determinant of the matrix   s s . . . s s  1 2 n n+1   s s . . . s s   2 3 n+1 n+2    . . .. .   . . . .  evaluated with respect to the last row.     sn sn+1 . . . s2n−1 s2n      1 z . . . zn−1 zn

The expression for the vector (bn−1, . . . , b0) one can easily obtain by multiplying Q(z)  2n  P sk and zk and taking the terms which involve only positive degrees of the variable z. k=1 2. Suppose that (s1, . . . , s2n) is an n-tuple of complex numbers with det ∆ 6= 0. Since the determinant of the system S, up to a sign, coincide with det∆, the system S has a

unique solution which gives rise to a pair of polynomials P and Q. I claim that these

are relatively prime polynomials. Indeed, if P = UW and Q = UV with V ∈ C[z] a

monic polynomial, then for any polynomial monic polynomial A ∈ C[z] of degree at most n − degV the coefficients of the polynomials AW and AV satisfy the system S, and thus

we derive a contradiction.

The next proposition is a direct corollary from Theorems 2.2, 2.1 and deeper investigation

of the system (2.1). We will employ Corollary 2.5, which is an immediate consequence

of Proposition 2.4, in the section 2.2.3. In the proposition and the corollary below we

denote the matrix ∆ constructed from the first 2n numbers s1, . . . , s2n by ∆n.

Proposition 2.4. Let s1, . . . , s2N be complex numbers, and suppose that ∆n 6= 0, where

n < N. Then the rational function F constructed from the 2n-tuple s1, . . . , s2n in Theo- 2N P sk −2N rem 2.2 satisfy the identity F = zk + o(z ) in a neighborhood of infinity if and only k=1 if det ∆n+1 = ... = det ∆N = 0. Chapter 2. Rational functions, moment-type problems, applications 12

An immediate corollary of the proposition above is

∞ Corollary 2.5 (Kronecker criterion). Let {sk}k=1 be an infinite sequence of complex

numbers and suppose that ∆n 6= 0, where n < N. Then the rational function F con- ∞ P sk structed from the 2n-tuple s1, . . . , s2n in Theorem 2.2 satisfy the identity F = zk in a k=1 neighborhood of infinity if and only if det ∆ν = 0 for ν > n.

Proof of Proposition 2.4. Let F = P/Q with P and Q relatively prime of degrees (n − 1) 2N P sk −2N and n, respectively, and Q monic. Suppose that F = zk + o(z ) in a neighborhood k=1 of infinity. Since P and Q relatively prime Theorem 2.2 implies that det ∆ν = 0 for n + 1 ≤ ν ≤ N.

To prove the converse, suppose that s1, . . . , s2N are complex numbers so that det ∆n 6=

N−1 0 and det ∆ν = 0 for n + 1 ≤ ν ≤ N, where n ∈ N. Let P = bN−1z + ... + b0

N−1 and Q = aN−1z + ... + a0. Expanding brackets in the identity P (z) = Q(z)·  2N  P sk −2N −N −1 0 1 N−1 zk + o(z ) and collecting coefficients before z , . . . , z , z , z , . . . , z , we k=1 obtain a system of linear homogeneous equations with (aN−1, . . . , a0) and (bN−1, . . . , b0) unknowns. It is the homogeneous system of linear equations corresponding to the sys- tem (2.1):   ∆ 0  N    x = 0, C −E where x is the vector (a0, . . . , aN−1, bN−1, . . . , b0), and C - is an N × N matrix. It is a straightforward verification, as in the beginning of the proof of Theorems 2.1 and

2.2, that any (non-zero) solution of the system above gives rise to polynomials P =

N−2 N−1 bN−2z + ... + b0 (bN−1 is automatically zero) and Q = aN−1z + ... + a0 with the 2N P sk −2N quotient F = P/Q equal to zk + o(z ) in a neighborhood of infinity. k=1

Since det ∆N = 0, the system above has a non-zero solution. Consider then a function 2N P sk −2N F = P/Q equal to zk + o(z ) in a neighborhood of infinity. By Theorems 2.1, the k=1 function F is a quotient of two relatively prime polynomials of degree at most n. The Chapter 2. Rational functions, moment-type problems, applications 13

application of Theorem 2.2, shows that F coincide with the rational function constructed from the 2n-tuple as in Theorem 2.2.

The following two problems, similar to problems in the theory of moments, are closely

related to our discussion.

Problem A. Given the first 2n moments sk (0 ≤ k ≤ 2n − 1) of the system of non-zero

(complex-valued) masses ρ1, . . . , ρn located at n pair-wise distinct points z1, . . . , zn on n P k−1 the complex line, reconstruct the points and the masses. Here sk = ρizi . i=1 Problem B. Given a sequence of 2n numbers. Is there a system M of n non-zero masses

at some n pair-wise distinct points on the complex line such that s1, . . . , s2n is a sequence of the first 2n moments of the system M?

Theorem 2.6. 1. A sequence s1, . . . , s2n is a moment sequence of a system M if and only if the polynomial Q = det ∆(z) has no multiple roots and det ∆ 6= 0.

2. If the polynomial Q satisfies the above condition, we recover M in the following

way: take the 1-form ω = Rdz with R the rational function constructed from the

sequence s1, . . . , s2n as in Theorem 2.3. Then, points of M correspond to the finite poles of the form ω (which are the roots of Q), masses - to the residues of ω.

The next lemma shows the connection of Problems A and B with Theorem 2.1.

Lemma 2.7. For pair-wise distinct complex numbers zk and non-zero ρk consider the n rational function F = P ρk . The following is valid z−zk k=1

∞ X sk 1. Its Taylor expansion at infinity is given by the formula F = . zk+1 k=0 2. The function F has the representation as a quotient S/T of relatively prime poly-

nomials S and T with S ∈ C[z] a polynomial of degree at most n − 1 and T =

(z − z1) ... (z − zn). Chapter 2. Rational functions, moment-type problems, applications 14

a a Proof of Lemma 2.7. 1. Indeed, for any complex a, b the function z−b = b = z(1− z ) ∞ ∞ a P b k P abk z ( ( z ) ) = zk+1 for sufficiently large |z|. Applying this identity to each summand k=0 k=0 ρk of the function F , we complete the proof of the first part. z−zk

2. Reducing F to the common denominator, we obtain F = S/T , where S is a

polynomial of degree at most n − 1 and T = (z − z1) ... (z − zn). Since ρk were non-zero numbers, the polynomials S and T are relatively prime.

Proof of Theorem 2.6. 1. Assume that s1, . . . , s2n is a moment sequence of n non-zero n P ρk S masses ρ1, . . . , ρn at points z1, . . . , zn. Consider the function F = = , where S z−zk T k=1 is a polynomial of degree at most n − 1, T = (z − z1) ... (z − zn); S and T are relatively

det ∆(z) prime. According to Theorem 2.1, det ∆ 6= 0 and (z − z1) ... (z − zn) = det ∆ .

det ∆(z) 2. Given that a polynomial Q = det ∆ has no multiple roots and det ∆ 6= 0, we consider a 1-form ω = Rdz, where R is a rational function of degree n constructed from

the sequence s1, . . . , s2n according to Theorem 2.3. We consider a partial fraction decom- position of the function R. According to the formulas of Theorem 2.1, the denominator n of the function R is up to a multiple det ∆ equal to the polynomial Q. Thus R = P ρk , z−zk k=1 where zk are roots of the polynomial Q and ρk are some complex numbers. Since R is a

rational function of degree n, we note that none of the numbers ρk is equal to zero.

Consider a system of masses ρk located at points zk. According to Lemma 2.7, the first 2n moments of this system coincide with the first 2n coefficients of the Taylor expansion

of the function R at infinity, which by the construction are equal to s1, . . . , s2n. Chapter 2. Rational functions, moment-type problems, applications 15

2.2 Applications

2.2.1 Structure of finite mappings

We proceed with applications of formulas given in Theorems 2.3, 2.6. First, we prove a

very general theorem about the structure of a finite degree mapping between two sets

(Theorem 2.8), then we prove its topological counterpart for a finite multiplicity mapping

between two topological spaces (Theorem 2.13).

Let A1,...,A2d be complex-valued functions on a set X. We denote by

    A1(p) A2(p) ...Ad+1(p)   A1(p) A2(p) ...Ad+1(p)     A (p) A (p) ...A (p)    2 3 d+2  A2(p) A3(p) ...Ad+2(p)  . . .  ∆(p) =   , ∆(p, z) =  ......   . . . .   . . .   . . .. .          Ad(p) Ad+1(p) ...A2d(p)    Ad(p) Ad+1(p) ...A2d(p)   1 z . . . zd where p ∈ X.

We shall say that a mapping F : M 7→ N between two sets M and N is a degree d mapping, if any element from N has exactly d preimages. For any complex-valued function h on the set M we define a function traceF h on the set N. On an element d P p ∈ N its value is equal to h(pk), where the summation goes over all preimages k=1 p1, . . . , pd of the element p.

Let F0,...,Fd be complex-valued functions on the set N. Denote by M(F0,...,Fd) a subset of a product N × C, given by the following condition: (p, z0) ∈ N × C, if

d Fd(p)z0 + ... + F0(p) = 0. The projection of the product N × C onto its first component induces a mapping π from the set M(F0,...,Fd) to N. Suppose that for any point p ∈ N

d a polynomial Fd(p)z + ... + F0(p) = 0 has exactly d pair-wise distinct complex roots.

Then the degree of the mapping π : M(F0,...,Fd) 7→ N is d. Next theorem shows that under some additional assumptions any mapping of degree d has the mentioned above Chapter 2. Rational functions, moment-type problems, applications 16

form and, more importantly, functions Fk could be chosen in a universal form, which will be convenient for the proof of a converse of Abel’s theorem.

Theorem 2.8. Let f : M 7→ N be a mapping of degree d; ρ, g - are complex-valued functions on M. Assume that the function ρ does not vanish on M and a mapping

k F = (f, g): M 7→ N × C is injective. Let A = (A0,...,A2d−1), where Ak = tracef g ρ.

Then there are universal (depending on d only) polynomials S0,...,Sd,T0,...,T2d−1,T2d in 2d variables with integer coefficients such that:

1. F (M) = M(S0(A),...,Sd(A)), where A = (A0,...,A2d−1), and the following dia- gram commutes: F M / M(S0(A),...,Sd(A)) @ @@ π mmm @@ mmm f @@ mmm @ mmm N vmm

2. ρ = f ∗( T2d−1 (A))g2d−1 + f ∗( T2d−2 (A))g2d−2 + ... + f ∗( T0 (A)). T2d T2d T2d

Remark 2.2. Theorem 2.8 can easily be extended to functions f and g taking values in

an arbitrary field K. Moreover, the universal polynomials S0,S1,...,Sd,T0,T1,...,Td

remain unchanged. The same polynomials are valid for any field K. We will employ this remark in section 2.2.2.

Lemma 2.9. For any two relatively prime polynomials with complex coefficients P =

n m ˜ ˜ any + ... + a0 and Q = bmy + ... + b0, there are unique polynomials P and Q such that P Q˜ + QP˜ = 1 with deg(P˜) < n, deg(Q˜) < m. The coefficients of P˜ (resp. Q˜) are

˜ Rk Rk ˜ expressed in the form ak = Res(P,Q) (resp. bk = Res(P,Q) ), where Rk (Rk) are universal (depending on d only) polynomials in 2d variables with integer coefficients and Res(P,Q)

is the resultant of the polynomials P and Q.

−1 Proof of Theorem 2.8. 1. Let p1, . . . , pd be the preimages f of an element p ∈ N.

Consider the masses ρ(p1), . . . , ρ(pd) located at the points g(p1), . . . , g(pd) According to

Theorem 2.2, the polynomial Qp(z) = det∆(p, z) has exactly d roots – g(p1), . . . , g(pd). Chapter 2. Rational functions, moment-type problems, applications 17

It implies that a) F (M) = M(S0(A),...,Sd(A)) with Sk being a (n + 1, k + 1) minor of the matrix det∆(p, z); b) the commutativity of the diagram.

2. For any element p ∈ N we construct a 1-form R dz = Pp dz such that Res R dz = p Qp g(pk) p 0 ρ(pk), where k = 1, . . . , d. It is possible due to Theorem 2.2. The polynomials Qp and Qp

˜ ˜0 ˜0 0 ˜ are relatively prime, therefore we can find polynomials Qp and Q p with QpQ p+QpQp = 1.

0 ˜ Pp ˜ In particular, Q (g(pk))Pp(g(pk)) = 1. Now, Resg(p )Rpdz = 0 |g(p ) = PpQp(g(pk)). As p k Qp k

a consequence of Theorem 2.1 the functions bn−1, . . . , b0, whose values at the element

p ∈ N are the coefficients of the polynomial Pp, and functions an, . . . , a0, whose values at

the element p ∈ N are the coefficients of the polynomial Qp, are related by the formula:       bn−1 A1 0 0 ... 0 an             bn−2 A2 A1 0 ... 0  an−1   =      .   . . . . .   .   .   ......   .              b0 An A3 A2 ...A1 a1 ˜ Combining the result Lemma 2.9 regarding coefficients of the polynomial Qp and the above formula we complete the proof.

Proof of Lemma 2.9. Denote by Vk a (k + 1)-dimensional vector space of polynomials

with complex coefficients of degree at most k. Consider a mapping φ : Vn × Vm 7→ Vm+n defined as follows: φ((S,T )) = PS+QT , where (S,T ) ∈ Vn×Vm. This is a linear mapping between the same dimension vector spaces. The kernel of the mapping φ is trivial, since polynomials P and Q are relatively prime. Thus there are unique polynomials P,˜ Q˜ of degree at most m − 1 and n − 1 respectively such that P Q˜ + QP˜ = 1. Choose a basis

n m m+n (1, 0), (y, 0) ..., (y , 0), (0, 1), (0, y),..., (0, y ) of Vn × Vm and a basis 1, y, . . . , y of

Vm. The (m+n)×(m+n) matrix Φ of the mapping φ in the above basis is the transpose of the Sylvester matrix: Chapter 2. Rational functions, moment-type problems, applications 18

  a a . . . a 0 0 0 0 ... 0  0 1 n     0 a a . . . a 0 0 0 ... 0  0 1 n     ......   ......       0 0 . . . a0 a1 . . . an 0 ... 0     b b . . . b 0 ... 0 0 ... 0  0 1 m       0 b0 b1 . . . bm 0 ... 0 0 ... 0    ......   ......      0 0 . . . b0 b1 . . . bm 0 ... 0

˜ ˜ −1 Coefficients of P (Q) are the first n components (last m components) of the vector Φ e1,

−1 where e1 is the vector (1, 0,..., 0). The elements of the inverse matrix Φ are up to a multiple detΦ given by the universal polynomials. Finally, the determinant of the matrix

Φ is, by definition, is the resultant of polynomials P and Q.

Next we are going to prove an analogue of Theorem 2.8 in the category of topological

spaces. With a symmetric polynomial T (z1, . . . , zd) and a continuous function h : M 7→ C

(d) (d) on the topological space M one can associate a function hT : M 7→ C defined as

(d) (d) follows: hT (x) = T (h(x1), . . . , h(xd)), where x ∈ M is a d-tuple (x1, . . . , xd).

Proposition 2.10. The function h(d) is continuous.

Proof. We can think of h(d) as a function on M d = M × ... × M. As a function on M d it is: 1) invariant under the natural action of the permutation group Sd; 2) continuous as being a composition of continuous mappings hd : M d 7→ Cd and T : Cd 7→ C. Here

d (d) h (x1, . . . , xd) = (h(x1), . . . , h(xd)). Thus, by definition, the function h is continuous on M (d)

Below we assume topological spaces to be not very “bad”(for example, metrizable

topological spaces). We shall say that a continues mapping F : M 7→ N between two

topological spaces M and N is a branch covering of a degree d if: Chapter 2. Rational functions, moment-type problems, applications 19

1) there exists an open everywhere dense subset U ⊂ N such, that F : F −1(U) 7→ U is a covering of degree d;

2) there exists a continues mapping H : N 7→ M (d) - the dth symmetric power of M,

H such that the following diagram commutes: N / (d) C M CC F (d) ww CC ww diag CC ww C! {ww N (d) Here diag is a diagonal mapping; F (d) is a mapping naturally induced by the F : the

(d) (d) image of a d-tuple (x1, . . . xd) ∈ M is defined as d-tuple (F (x1),...F (xd)) ∈ N . Due to the commutativity of the diagram in 2), the mapping H is an inclusion and

the preimage F −1(p) of any point p ∈ N is an element H(p) ∈ M (d). Thus we can think

−1 of F (p) as a collection of points (p1, . . . , pl) with multiplicities (a1, . . . , al), such that l P ak = d. We shall write that multF pk = ak. Since an arbitrary point a ∈ M belong to k=1 F −1(F (a)) the multiplicity is well-defined for all points of the topological space M .

We recall that a continuous mapping between topological spaces is open if the image

of an open set is open. It is a straightforward exercise to check the following

Proposition 2.11. Let F : M 7→ N be a continuous mapping between two topological

spaces such that F a finite covering over an open everywhere dense subset U ⊂ N, and

each point in N has a finite number of preimages. Then F : M 7→ N is a branch covering

if and only if F is proper and open.

Let F : M 7→ N be a branch covering of degree d. The trace of a continuous function

∗ (d) h : M 7→ C under the mapping F is H (hT ), where T = z1 + ... + zn. Notation: traceF h.

Remark 2.3. It easily follows from the definition that on the set U ⊂ N the trace of

−1 the function h coincide with the traceF |F −1(U)h, where the mapping F : F (U) 7→ U is considered as the degree d mapping between two sets.

∗ (d) Proposition 2.12. For any symmetric polynomial T the function H (hT ) is continuous. Chapter 2. Rational functions, moment-type problems, applications 20

(d) Proof. Indeed, it is a push-back of the continuous function hT under the continuous mapping H.

Theorem 2.13. Suppose that M ⊂ N × C is a topological subspace and the projection

π : N × C 7→ N onto the first multiplier induces a branch covering π : M 7→ N of degree d. Let ρ be a complex-valued, continuous function on M that vanishes on nowhere dense

subset of M. Then on the complex line C with a fixed coordinate z there is a family of rational 1-forms ω dz = Pp dz parameterized by points of N such that: p Qp

1. For any p ∈ N, the zeros of the polynomial Qp counted with multiplicities are in one-to-one correspondence with the points of F −1(p) counted with multiplicities; if

−1 A ∈ F (p) then ResAωp = multF A ρ(A).

d−1 2. The polynomials Pp and Qp has the form Pp = bd−1(p)z + ... + b0(p) and

d d−1 Qp = z + ad−1(p)z + ... + a0(p) with ak, bk : N 7→ C continuous functions. The set V = {p ∈ N| det∆(p) 6= 0} is open and everywhere dense in N. Moreover, if

det∆(p,z) k p ∈ V , then the following formulas are valid: Qp = det∆(p) , where Ak = traceF z ρ and z is considered as a function on M, and       bd−1 A1 0 0 ... 0 1             bd−2 A2 A1 0 ... 0  ad−1   =     .  .   . . . . .   .   .   ......   .              b0 Ad Ad−1 A2 ...A1 a1

Proof of the Theorem 2.13. Choose an everywhere dense subset V ∈ N such that ρ does not vanish on F −1(V ) and F : F −1(V ) 7→ V is a degree d mapping.

Now we will complete the proof in two steps. In the step one we prove Theorem 2.8(1) for the points p ∈ V , and we establish Theorem 2.8(2); in the step two we verify Theo- rem 2.8(1) for all the points p ∈ N.

−1 Step 1. Note that at points of F (V ) the multiplicities are equal to 1. Let p1, . . . , pd be the preimages of a point p ∈ V ⊂ N. The formulas in the first and second parts of the Chapter 2. Rational functions, moment-type problems, applications 21

theorem directly follow from Theorem 2.2 applied to the masses ρ(p1), . . . , ρ(pd) located at the points z(p1), . . . z(pd).

We now verify the continuity of the coefficients ak and bk. The functionz ˜ : M 7→ C is a pull-back of a coordinate function z under the projection N × C 7→ C, and thus is continuous. Any symmetric polynomial T in d variables gives rise to a continuous

(d) (d) ∗ (d) functionz ˜T on M . According to Vieta’s formulas, ak = H (˜zT ), where T is the k-th principal symmetric polynomial and k = 0, . . . , d − 1. Thus both ak and bk coincide with continuous functions defined on the whole N. We also conclude that the formula for the coefficients bk in terms of ak, in fact, is valid for all p ∈ N due to the continuity of functions involved.

−1 Step 2. Take a point p ∈ N\V . Its preimage F = H(p) = (c1p1, . . . , clpl), where

∗ (d) ck are multiplicities. In Step 1 we found that ak = H (˜zT ), therefore, according to the inverse of Vieta’s formula, complex numbersz ˜(p1),..., z˜(pl) are the roots of the

d d−1 polynomial Qp = z +ad−7(p)z +...+a0(p) of the multiplicities c1, . . . , cl respectively.

Finally, let us verify the formula for the residue of ωp, say, at a pointz ˜(p1). Choose a sequence {rk} of points of V that approaches p as k → ∞. Note that H(rk) → H(p) and therefore, for a sufficiently large k, points of H(rk) are divided into l groups of cl points, respectively. Each group of points converges to the corresponding pl as k → ∞. d P rk P ρ(rkm) −1 Now taking the limit of both side in the identity : = , where F (rk) = Qr z−z˜(rkm) k m=1 (rk1, . . . , rkd) we obtain the expected formula for the residue.

Under the assumptions and notations of the previous theorem the following is the straight-

forward corollary from the properties of the forms ωpdz in Theorem 2.8(2).

Corollary 2.14. The topological subspace M ⊂ N × C is the zero locus of the pseudo-

d d−1 polynomial Q = z + ad−1z + ... + a0 with coefficients continuous, complex-valued functions on N. On an open everywhere dense subset V × C in N × C the pseudo- det∆(p,z) polynomial Q(p) = det∆(p) . Chapter 2. Rational functions, moment-type problems, applications 22

˜ k ρ Assume additionally, that the functions Ak = traceF z 0 well-defined on the subset Qz V in N, admit continuous extension to the whole N. The next is the straightforward corollary from the properties of the forms ωpdz in Theorem 2.8(1).

d−1 Corollary 2.15. There is a pseudo-polynomial P = bd−1+z +...+b0 whose coefficients are continuous functions on N so that the function ρ coincides with the restriction of the       ˜ bd−1 A1 0 0 ... 0 1          ˜ ˜    bd−2 A2 A1 0 ... 0  ad−1 pseudo-polynomial to M. Moreover,   =     .  .   . . . . .   .   .   ......   .              ˜ ˜ ˜ ˜ b0 Ad Ad−1 A2 ... A1 a1

Remark 2.4. The statement of the corollary 2.15 is a sort of Weierstrass’ preparation

theorem in the category of topological spaces.

We can think of the topological subspace M ⊂ N × C from the theorem 2.13 as a

hypersurface in N × C. The next proposition, which for simplicity we state for cover- ings, locally reduces the case of an arbitrary codimension subspace M to the case of a

hypersurface.

Let K be either R or C. A polynomial in n variables whose coefficients are continu-

ous, K-valued functions on N is called a pseudo-polynomial. We say that P1,...,Pk ∈

n C(N)[x1, . . . , xn] define a complete intersection in N × K if, at each common zero of the

pseudo-polynomials, the Jacobian matrix of P1,...,Pk with respect to variables x1, . . . , xn has the maximal rank.

Proposition 2.16. Suppose that M ⊂ N ×Kn is a topological subspace and the projection

π : N ×Kn 7→ N onto the first multiplier induces a covering π : M 7→ N of degree d. Then for a sufficiently small neighborhood U of any prescribed point in N, there are coordinates

n ∗ T −1 y1, . . . , yn ∈ (K ) so that the projections πj = (π, yj): M π (U) 7→ U ×K are one-to-

one. In addition, if ρ : M 7→ K is a continuous function on M that does not vanish on T n n an open everywhere dense subset of M, then M (U × K ) in N × K is a subset of the Chapter 2. Rational functions, moment-type problems, applications 23

complete intersection defined by the pseudo-polynomials P1,...,Pn. Each Pj is obtained

from the formula in Corollary 2.14 by replacing z (including the functions Ak) with yj.

−1 Proof. Let p1, . . . , pd be the preimages π (p) of a point p ∈ N. For a basis l = (`1, . . . , `n) in (Kn)∗ to satisfy the condition of the proposition it is necessary and sufficient that

`i(pk) 6= `j(pk), for every 1 ≤ i 6= j ≤ n, 1 ≤ k ≤ d, and any point p ∈ N. Choose such a basis l. By continuity, the same inequalities are valid for preimages of points in a sufficiently small neighborhood Up of p ∈ N.

T −1 Corollary 2.14 applied to the degree d covering between πj(M π (U)) ⊂ U × K and U induced by the projections of U ×K onto the first multiplier, ensures the existence of the pseudo polynomials P1 ∈ C(U)[y1],...,Pn ∈ C(U)[yn] so that the equation Pj = 0 T −1 defines πj(M π (U)) in U×K. The Jacobian of P1,...,Pn with respect to the variables

∂P1 ∂Pn y1, . . . , yn at a point A ∈ {P1 = ... = Pn = 0} is ... evaluated at A. Now, each ∂y1 ∂yn

multiplier in the latter expression does not vanish since each polynomial Pj(π(A)) ∈ K[yj] has no multiple roots.

2.2.2 Moment-type problem and Galois theory

The aim of this section is to prove Proposition 2.18 which we heavily use in the prove

of the Converse of Abel’s theorem - rational case. Proposition 2.18 has the flavor of

the differential Galois theory. As a warm up we prove Proposition 2.17 which one may

consider as the key assertion in the main theorem in Galois theory. We refer to the course

of Khovanskii at the University of Toronto where this view-point was implemented.

Let G be a finite group acting on a field K with K0 ⊂ K the field of invariants.

Suppose that α and β are non-zero elements of the field K. We say that β is subordinate to α if g(α) = α for some g ∈ G implies that g(β) = β.

Proposition 2.17. The element β is subordinate to α if and only if β = P (α) with Chapter 2. Rational functions, moment-type problems, applications 24

P ∈ K0[y] a polynomial.

Proof of Proposition 2.17. If β = P (α) with P ∈ K0[y] then β is subordinate to α since

the field K0 is element-wise invariant under the action of G.

Conversely, let y1, . . . , yd be the orbit of α under the action of G with y1 = α. Define

non-zero K-valued masses ρ1, . . . , ρd at y1, . . . , yd by the following rule: if yk = g(y1),

then ρk = g(β). The element g that takes y1 to yk is not unique and is defined up to

the left multiplication by the stabilizer of y1, and thus, by the subordinate property of

the element β, the mass ρk is well-defined. In particular, ρ1 = β. Consider the first 2n d P k−1 moments sk = ρizi with 0 ≤ k ≤ 2d − 1 of the system of masses. Note that each sk i=1 is invariant under the action of G, and thus sk ∈ K0.

Theorem 2.8(2) (see Remark 2.2) with M = {y1, . . . , yn} and N being just one point,

the K-valued functions g and ρ being the collection of y1, . . . , yd and ρ1, . . . , ρn, respec-

tively, shows that there is a polynomial P ∈ K0[y] of degree at most 2d − 2 so that

ρ = P (g). In particular, substituting y1 ∈ M, we obtain β = P (α). This completes the proof.

We now proceed with the main theorem in this section. Let G be an (infinite) group acting on a field K with K0 ⊂ K the field of invariants.

Proposition 2.18. Let L be a field and K ⊂ L. Suppose that ρ1, . . . , ρd are non-zero

L-valued masses at pairwise distinct elements y1, . . . , yd of L so that the first 4d moments d P k−1 sk = ρizi belong to K and satisfy the following condition: (gsk − sk) ∈ K0 with 0 ≤ i=1 k ≤ 4d−1 and g ∈ G arbitrary. Then the coefficients of the polynomial (y−y1) ··· (y−yd) are in K0.

Proof of Proposition 2.18. We follow the notations in Theorem 2.1. Let K(y) be the

field of rational functions with coefficients in K. Let L¯ be the algebraic closure of L. By 2d P sk −2d Lemma 2.7, there is the rational function R = P/Q = yk + o(y ) as in Theorem 2.2 k=1 Chapter 2. Rational functions, moment-type problems, applications 25

4d ρ1 ρd P sk −4d and R = +...+ . Note that R = k +o(y ). Fix g ∈ G. Since g : 7→ is y−y1 y−yd y K K k=1 an automorphism, the determinant constructed from the sequence gs0, . . . , gs2n−1 is equal 2d P gsk −2d to g det ∆ 6= 0, and thus there is the rational function R1 = P1/Q1 = yk + o(y ). k=1 By the formulas in Theorem 2.1, the coefficients of polynomials P and Q belong to the

field K and the polynomials P1 and Q1 are obtained by applying g to the coefficients P 4d P gsk −4d and Q, respectively. We write P1 = gP and Q1 = gQ. Thus R1 = yk + o(y ) and k=1

4d X gsk − sk R − R = + o(y−4d). 1 yk k=1

The function R1 − R ∈ K(y) has degree at most 2d. Again, Theorem 2.1 implies that

R1 − R ∈ K0(y). The discriminant discr(Q) of Q is a certain “universal” polynomial with in the co- efficients of Q. We conclude that discr(Q1) = gdiscr(Q). Since Q1 is monic, it has no

µ1 µd multiple roots. We conclude that R1 = +...+ with w1, . . . , wd pairwise distinct y−w1 y−wd ¯ ¯ elements in L and µ1, . . . , µd ∈ L non-zero. Let R1 − R = P0/Q0, where P0,Q0 ∈ K0[y] are relatively prime polynomials. We then have the following two options: for each

i = 1, . . . , d, either yi = wj for some j = 1, . . . , d or yi is algebraic over the field K0 and

is a zero of the polynomial Q0, of degree at most 2d. Let A be the finite set that contains

the elements y1, . . . , yd and all the algebraic conjugated to the element yk if some yk ∈ K

is algebraic over the field K0. We conclude the new d elements w1, . . . , wd, the roots of the monic polynomial gQ, belong to the set A. ¯ Now consider µ1, . . . , µd as non-zero L-valued masses at pairwise distinct elements d ¯ P k−1 w1, . . . , wd of L. Then the first 4d moments of the new system µiwi are equal to i=1 2 gsk, which belong to K and g sk −gsk = g(gsk −sk) = (gsk −sk) ∈ K0 for 0 ≤ k ≤ 4d−1. Apply the same argument to the new system of masses. We find that yet another d

(1) (1) 2 elements w1 , . . . , wd , the roots of the monic polynomial g Q, belong to the finite set A.

Since A is a finite set, it follows that, there is a non-negative integer m so that the Chapter 2. Rational functions, moment-type problems, applications 26

polynomial gmQ, obtained from Q by applying g to its coefficients m times, coincide with

Q.

Now Q = (y − y1) ··· (y − yd) = det ∆(y) / det ∆. Suppose that gsk − sk = pk ∈ K0,

n then g sk − sk = npk for any integer n. Consider the function

  s + tp s + tp . . . s + tp  0 0 1 1 d d     s + tp s + tp . . . s + tp   1 1 2 2 d+1 d+1     . . .. .  Q(t) = det  . . . .  .     sd−1 + tpd−1 sd + tpd . . . s2d−1 + tp2d−1     1 y . . . yd

d We write Q(t) = ∆d(t)y + ... + ∆0(t). For each k, the functions ∆k(t)/∆d(t) are rational in t with coefficients in K and for any integer t, ∆k(t)/∆d(t) coincide with

t the polynomial g Q. Thus ∆k(tm)/∆d(tm) is identically equal to the corresponding coefficient of Q for any integer t. Since ∆k(t)/∆d(t) is rational in the variable t, we conclude ∆k(1)/∆d(1) = ∆k/∆d. We just showed that the coefficients of the polynomial Q are invariant under the action of the element g ∈ G. Since g could be chosen arbitrary, we conclude that Q ∈ K0[y].

2.2.3 A GAGA-type theorem

The aim of this appendix is to prove the following

n Theorem 2.19. Let v1, . . . , vn be linearly independent vectors in C . Suppose that f :

U 7→ C is an analytic function on a connected domain U ⊂ Cn such that for any line l

parallel to one of the directions vk the restriction f|l T U is a rational function. Then f coincide with a rational function in n complex variables.

Theorem 2.19 is a straightforward corollary of Propositions 2.20, 2.21 below. We remind that, if a rational function R is the quotient of two relatively prime polynomials P and Chapter 2. Rational functions, moment-type problems, applications 27

Q in one complex variable, then the maximum of the degrees of P and Q is called the

degree of the rational function R.

Proposition 2.20. Let f : U 7→ C be an analytic function defined on a connected domain

n U in C × C . Suppose that that for any horizontal line l the restriction of f|l T U is a rational function. Then the degrees of all the rational functions are uniformly bounded above.

Let K be either R or C.

n Proposition 2.21. Let v1, . . . , vn be linearly independent vectors in K . Suppose that f is a K-valued function on a connected domain U ⊂ Kn such that for any line l parallel to one of the directions vk the restriction f|l T U is a rational function of degree at most nf , where nf does not depend on the line l. Then f coincide with a rational function in n variables of degree at most nf .

For the proof of Proposition 2.20 we need the following well-known criterion, which easily

∞ follows from Corollary 2.5. Let {sk}k=0 be an infinite sequence of complex numbers. We denote the matrix ∆ constructed on the page 9 from (2n − 1) numbers s1, . . . , s2n−1 by

∆n. Then

Proposition 2.22 (Kronecker criterion). Let n ∈ N. There is a rational function F ∈ ∞ P k C(z) of degree n so that F = skz in a neighborhood of the origin if and only if k=0 det ∆n 6= 0 and det ∆ν = 0 for ν > n.

Proof of Proposition 2.22. Indeed, it immediately follows from Corollary 2.5 after send- ing the origin to the infinity by means of the substitution z = 1/w.

Proof of Proposition 2.20. Choose a closed disk V ⊂ Cn and a closed disk ∆ ⊂ C so that ∆ × V ⊂ U. In ∆ × V the function f admits the following representation f =

2 c0 + c1y + c2y + ..., where y is a coordinate on the line C and ck are holomorphic functions on V . Since ck are kth partial derivatives of the function F with respect to the Chapter 2. Rational functions, moment-type problems, applications 28

variable y, if ck = 0 on V for k ≥ 1, then all the partial derivatives of F with respect to y vanish on the domain U and the degrees of all the rational functions f|l T U are bounded above by, say, 1.

Otherwise, let Sm ⊂ V be the set of points p so that det ∆m 6= 0 and det ∆ν = 0 for

∞ ν > m, where the matrices ∆k are constructed from the sequence {sk}k=0.

The sets Sm are closed and their union is V . Since V is a Baire category space, there is an integer N and a point p ∈ V so that the set SN contains a small neighborhood of

p ∈ V . Thus all the functions det ∆ν with ν > N vanish on that neighborhood. The

entries of all the matrices ∆ν are certain partial derivatives of the function F with respect

to y. Since U is connected, we conclude that the functions det ∆ν are identically zero on

U if l ≥ N. Proposition 2.5 ensures that the degrees of all the rational functions f|l T U are bounded above by N, and this completes the proof.

Proof of Proposition 2.21. Note, first, that the rational function F = P/Q given as a

quotient of two relatively prime polynomials P,Q ∈ K[y] of degrees at most n is uniquely

determined by its values at 2n + 1 pairwise distinct elements y1, . . . , y2n+1 of the field K

(assuming that the polynomial Q does not vanish at the elements yk). Indeed, assuming

that F and F1 attains the same values at y1, . . . , y2n+1, we find a rational function F −F1

of degree at most 2n that vanish at 2n+1 points y1, . . . , y2n+1. Thus F −F1 is identically equal to zero.

n n Now, if F = P/Q with P = bnz + ... + b0 and Q = anz + ... + a0, then the 2n + 1

conditions F (yk) = P/Q(yk), where (1 ≤ k ≤ 2n + 1), gives rise to a system S of homo-

geneous linear equations P (yk) = F (yk)Q(yk) on the coefficients (bn, . . . , b0, an, . . . , a0). Below is the (2n + 1) × (2n + 2) matrix of the system S: Chapter 2. Rational functions, moment-type problems, applications 29

  n n y1 . . . y1 1 −F (y1)y1 ... −F (y1)y1 −F (y1)    n n   y2 . . . y2 1 −F (y2)y2 ... −F (y2)y2 −F (y2)    . (2.2)  . . . .   . . .. .      n n y2n+1 . . . y2n+1 1 −F (y2n+1)y2n+1 ... −F (y2n+1)y2n+1 −F (y2n+1)

Conversely, any solution of the system S gives rise to polynomials P˜ and Q˜ so that ˜ ˜ ˜ ˜ P (yk) = F (yk)Q(yk) for k = 1,..., 2n+1. It is then easily follows that the function P/Q coincide with F . We conclude that if one of the polynomials P and Q has degree n or, in other words, if the degree of the rational function F is n, then the matrix of the system

S has rank 2n + 1.

We return to the proof of the Proposition. Let (x, y) = (x1, . . . , xn−1, y) be an affine

n n−1 system of coordinates in C = C × C in which the vectors vk correspond to the coordinate axes and, particularly, vn correspond to the y-axis. Let ∆ ⊂ C be an interval n−1 ∼ with 2n + 1 fixed points y1, . . . , y2n+1 and V ⊂ C an open disk. Suppose that Up = ∆ × V ⊂ U is a neighborhood of a point p ∈ U such that:

1. For any a ∈ V , the function F restricted to the vertical line `a passing through the point a ∈ V is a rational function of degree precisely n.

2. Some (2n + 1) × (2n + 1) minor of the matrix (2.2) constructed with F = F |`a is not zero for any a ∈ V .

n bnz +...+b0 Let F |` = n , where ak and bk depends on a ∈ V and are defined up to a anz +...+a0 multiplications by a non-zero element of K. The condition 2 above ensures that one of

the coefficients bn, . . . , b0, an, . . . , a0 is not zero for any a ∈ V . Without loss of generality

n bnz +...+b0 we assume that an 6= 0. Then F |` = n n−1 , where bn, . . . , b0, an−1, . . . , a0 are a z +an−1z +...+a0 some functions on V which can be explicitly found by applying the Cramer’s rule to the

system (2.2) with F taken to be F |`a . By the induction we may assume that the functions Chapter 2. Rational functions, moment-type problems, applications 30

F (y1, x),...,F (y2n+1, x) are rational in the variables x1, . . . , xn−1. Since F (y1, a) = F |`a , we conclude that bn, . . . , b0, an−1, . . . , a0 are rational function in the variables x1, . . . , xn−1.

We complete the proof by showing the existence of a domain Up with the properties

1, 2. Fix any point in the domain U ⊂ Cn. Choose a open disk V ⊂ Cn and an open disk

∆ ⊂ C so that ∆ × V ⊂ U. In ∆ × V the function f admits the following representation

2 f = c0 + c1y + c2y + ..., where y is a coordinate on the line R and ck are holomorphic functions on V . By Proposition 2.22, we can find n so that det ∆n is not identically zero near the fixed point but det ∆ν = 0 for any ν > n. Take p ∈ U near the fixed point so that det ∆n(p) 6= 0. Take a neighborhood of p ∈ U of the same type ∆1 × V1 but outside the hypersurface det ∆n = 0. Now fix 2n + 1 pairwise distinct points on ∆1. For a fixed

a ∈ V1 the rank of the matrix (2.2) with F = F |`a is 2n + 1, and thus there is a non-zero (2n + 1) × (2n + 1) minor of the matrix (2.2). By continuity, near the point a ∈ V the same minor does not vanish.

Proposition 2.19 allows us to show the following well-known result.

Corollary 2.23. Suppose that f is a meromorphic function on a projective space CP n.

Then f coincide with a rational function on CP n.

Proof of Corollary 2.23. For n = 1 it is a simple exercise in complex variables. Suppose

that n > 1. The polar locus Σ ⊂ CP n of the function f is an analytic subset of dimension

at most (n − 1). Choose an affine chart Cn ⊂ CP n so that the infinity hyperplane

n−1 n n CP∞ = CP \C does not belong to the polar locus Σ. Since the restriction of f to any

line l ( Σ is a rational function, it is sufficient to show that one can choose n linearly

independent vectors in Cn so that any line l parallel to one of the directions does not belong to Σ.

n−1 n−1 T This is done by choosing n points A1,...,An ⊂ CP∞ \(CP∞ Σ) with the linear

n−1 n span equal to CP∞ . In the affine chart C , each point Ak gives rise to a pencil of Chapter 2. Rational functions, moment-type problems, applications 31

parallel lines that pass through the point Ak at infinity. The intersection of any line ` from the pencil with Σ is an analytic subset of `, and since Ak ∈/ `, the line ` does not belong to Σ.

In the same spirit as corollary 2.23 one establishes the following result [8]

Proposition 2.24. Suppose that f is a meromorphic function in a neighborhood of a line in the projective space CP n. Then f coincide with a rational function on CP n. Chapter 3

Abel’s theorem

3.1 A version of Abel’s theorem for abstract and

plane curves

Let Γ be a compact connected Riemann surface and ω be a meromorphic form on it.

Consider a holomorphic non-constant mapping π :Γ 7→ CP 1. Outside of a finite set A ⊂

CP 1 the mapping π is a covering of a degree d. The preimage π−1(U) of a sufficiently small

1 neighborhood U of a point P ∈ CP \A is a disjoint union of open sets Uk, k = 1, . . . , d,

−1 where π : Uk 7→ U is an holomorpic isomorphism. Denote (π|Uk ) by gk. The trace form is by definition a 1-form on CP 1. In a sufficiently small neighborhood U of a point

P ∈ CP 1\A it is defined by the formula

d X ∗ traceπω = gkω, (3.1) k=1

Proposition 3.1 (Abel’s theorem for curves). The 1-form traceπω is a meromorphic form on CP 1. Moreover, if ω is a holomorphic 1-form, then the trace form is identically zero.

Proof. Indeed, the trace form is a meromorphic univalued 1-form on CP 1\A. Moreover,

32 Chapter 3. Abel’s theorem 33

since ω is meromorphic, traceπω has at most power growth when approaching points of the set A. That proves the first part of Proposition.

We proceed with the second part. From the definition of the trace form it follows that traceπω is holomorphic in the complement of the set A. We will verify that it remains regular near points of the set A. Take a point P ∈ A and consider any of its

preimages Q ∈ Γ. By appropriately choosing coordinates w and z near points Q and

P , respectively, we may assume that π(w) = z = wk with k a non-negative integer.

2 Let ω = ρ(w)dw = (a0 + a1w + a2w + ... +)dw be a coordinate expression of the

1 1 1 k k −1 form ω, where ρ(w) is a holomorphic function. Then w = z and dw = k z dz. P r n 1 Substituting these formulas, yields to traceπω = ( a˜nz )dz, where r = k + k − 1. Since P r the function ( a˜nz ) is univalued in a neighborhood of the point P ∈ A, the coefficients before all the fractional powers of z should be equal to zero. However, all the rational

n 1 P r numbers r = k + k − 1 are bigger then −1, and thus the function ( a˜nz ) is, in fact, holomorphic near the origin, which corresponds to P ∈ A. We conclude that the trace

form is holomorphic on the whole Riemann sphere. The application of the lemma below

completes the proof.

Lemma 3.2. There are no non-zero holomorphic 1-forms on the Riemann sphere or, in

other words, Ω1,0(CP 1) = 0. P 1,0 1 R Proof. Indeed, if ω ∈ Ω (CP ) then the function f(P ) = , where P is a point on the P0 Riemann sphere and the integration goes over any path that connects the fixed point P0 with the point P , is well-defined and holomorphic on CP 1. Therefore f = const and ω = df = 0.

Corollary 3.3 (Integral form of Abel’s theorem for curves). If γ1, . . . γd are preimages

of a path γ ⊂ CP 1\A under the mapping π :Γ 7→ CP 1, then d X Z ω = 0. k=1 γk Chapter 3. Abel’s theorem 34

R Proof. Indeed, the latter sum is equal to traceπω which is zero since the trace of the γ holomorphic 1-form ω is zero.

Remark 3.1. Originally Abel considered integrals of the kind R R(x, y(x))dx, where R ∈

R(x, y) is a rational function and y(x) is an algebraic function on an interval U ⊂ R. Usu- ally, an antiderivative of R(x, y(x)) is highly complicated and can not be expressed by a R R simple formula, however, as Abel showed, the sum R(x, y1(x))dx+...+ R(x, yn(x))dx of such antiderivatives over all branches of the function y(x) is always simple. Namely, P it is equal to Q(x) + ck log(x − zk), where Q is a rational function, and ck and zk are k some complex numbers. This is perfectly consistent with our discussion: on the Riemann

surface Γ of the algebraic function y(z) with z = x + iy one considers a meromorphic

1-form ω = Rdz. Then the above sum of integrals in the integral of tracezω, where

1 z :Γ 7→ CP . The 1-form tracezω is rational on the Riemann sphere, and thus its P antiderivative (restricted to the real axis) is of the type Q(x) + ck log(x − zk). k Let Γ ⊂ C2 be a smooth affine algebraic curve given as a zero locus of an irreducible

polynomial P ∈ C[x, y]. Additionally assume that the projective compactification Γ˜ of the curve Γ transversely intersects the line at infinity. The latter condition ensures that

the compactified curve Γ˜ ⊂ CP 2 is nonsingular. Throughout the manuscript, when there is no ambiguity, we will abuse the language by saying that a curve Γ transversely intersect

the line at infinity or by referring to points at infinity of the curve Γ.

Qdx∧dy Consider the 1-form dP on the curve Γ with Q a polynomial in two variables. The Qdx∧dy 2 definition of the form dP is local. In a C -neighborhood U of a point on the curve one searches for a 1-form τ such that the identity τ ∧ dP = Qdx ∧ dy is valid for all

the points of U T Γ. The form τ is not unique and is defined up to fdP , where f is a

holomorphic function on U (Proposition 3.17). Thus the form τ|Γ is well-defined. By

Qdx∧dy definition, the form dP is equal to τ|Γ. In coordinates:

0 1. If x is a local coordinate near a point a ∈ Γ, that is when Py(a) 6= 0, the form τ Chapter 3. Abel’s theorem 35

Qdx has the following coordinate expression τ = 0 . Py

0 2. If y is a local coordinate near a point a ∈ Γ, that is when Px(a) 6= 0, the form τ Qdy has the following coordinate expression τ = − 0 . Px    0 0    Qdx Qdy Pxdx + Pydy dP Notice that 0 − (− 0 ) = 0 0 = 0 0 = 0. Py Px Γ PxPy Γ PxPy Γ Qdx ∧ dy Conclusion: residue forms are holomorphic on the curve Γ. The following dP computation shows when residue forms are holomorphic on Γ˜ ⊂ CP 2. Let degQ = m. Consider a branch of the curve Γ at infinity. Without loss of generality we may assume

1 1 1 Qdx that u = is a local coordinate. Then x = , dx = − 2 du and 0 has a pole of order x u u Py at most m + 2 − (n − 1) = m − (n − 3) at u = 0. Thus if m ≤ n − 3 the residue form Qdx∧dy ˜ dP is holomorphic on the compactified curve Γ.

Proposition 3.4. Any holomorphic 1-form on the compactified curve Γ˜ is of the form

Qdx∧dy dP , where degP = n, degQ ≤ n − 3.

Qdx∧dy Proof. Indeed, it is not difficult to verify that residue forms dP , degQ ≤ degP −3 are linearly independent over C. Therefore the dimension of the vector space Ω that they constitute is equal to the dimension of polynomials in two variables of degree smaller

(n−1)(n−2) than degP − 2. The latter is 2 , where n = degP . On the other hand, it is well-known that the genus g(Γ)˜ of the curve Γ˜ is equal to

(n−1)(n−2) 1,0 ˜ ˜ 1,0 ˜ 2 and dimCΩ (Γ) = g(Γ), where Ω (Γ) is the space of holomorphic 1-forms on the curve Γ.˜ Two vector spaces Ω and Ω1,0(Γ)˜ have the same dimension and the first is a subspace of the second. Therefore they coincide.

As we observed, under some smoothness assumptions on the curve Γ, the residue forms Qdx∧dy ˜ dP , with degQ ≤ degP − 3, have intrinsic meaning on the projective closure Γ of Γ. The next proposition shows that they are ”almost” invariant under a projective

transformation. (For an invariant definition of a residue form see page 59). Chapter 3. Abel’s theorem 36

I recall that a projective transformation L is an element of P GL3(C). In the affine

2 2 2 l1 l2 chart C ⊂ CP with coordinates (x, y) the image of a point p ∈ C is L(p) = ( l (p)), l (p))

with l, l1, l2, ∈ C[x, y] polynomials of the first degree, which coefficients are the rows of

˜ degP l1 l2 ˜ degQ l1 l2 −1 the matrix L. Let P = l P ( l , l ) and Q = l Q( l , l ). The preimage L of Γ in the affine chart and the curve defined by the polynomial P˜ differs by a finitely many

points. Under these notations

Proposition 3.5. The following formula is valid

Qdx ∧ dy  lN−degQQ˜dx ∧ dy L∗ = , dP dP˜

where N = degP − 3.

We will use a simple lemma from linear algebra. Its multidimensional version we prove

in Lemma 3.21.

Lemma. For any three linear polynomials l1, l2, l3 ∈ C[x, y], the following identity is

valid: l1dl2 ∧ dl3 + l2dl3 ∧ dl1 + l3dl1 ∧ dl2 = det A dx ∧ dy, where the coefficients of the polynomials l1, l2, l3 respectively form the first three rows of the matrix A.

Qdx∧dy Proof of Proposition 3.5. By definition, dP = τ so that τ ∧ dP = Qdx ∧ dy. Ap- plication of L∗ to the both side yields to L∗(τ) ∧ L∗(dP ) = L∗(Qdx ∧ dy). Now

∗ l1 l2 P˜ dP˜ L (dP ) = dP ( l , l ) = d ldegP . The latter is equal ldegP when restricted to the curve defined by as polynomial P˜. ˜ ∗ dP˜ ∗ l1 l2 Q Substituting L (dP )|L−1(Γ) = ldegP ,L (Q) = Q( l , l ) = ldegQ , and using the identity

∗ l1 l2 det L dx∧dy from the lemma to compute L (dx ∧ dy) = d( l ) ∧ d( l ) = l3 , we obtain that L∗(τ) ∧ dP˜ = ldegP −3−degQ Q˜ dx ∧ dy for all the points of the curve defined by the polynomial P˜. This completes the proof.

Assume that M is a complex one-dimensional (not necessarily compact and connected)

manifold and ω is a holomorphic 1-form on it. Let N be a connected complex one-

dimensional manifold. We will say that a holomorphic mapping F : M 7→ N is finite Chapter 3. Abel’s theorem 37 if it satisfies two properties: 1) every point of the image N has a finite number of preimages, 2) there is a finite set A ⊂ N, which contains the set of critical values, such that every point of N\A has the same number of preimages. Then outside of the set A the mapping F is a finite covering and one can define traceF ω similarly to the formula (3.1) on the page 32. The number of preimages of a regular value we will call a degree of the mapping F . In case when N ⊂ C is a domain it will also be convenient to introduce a trace function [tracef ω]. If u is a fixed coordinate on the complex line C, the function

[tracef ω] is defined from the following identity:

tracef ω = [tracef ω]du.

f Proposition 3.6. Assume that f, g are holomorphic functions on M and R = g : M 7→ U is a finite mapping, where U ⊂ C is a domain. Assume that for every a ∈ U there exists a domain Ua ⊂ C which contains the origin such that the mapping πa = f −ag : Ma 7→ Ua,

−1 with Ma = πa (Ua), is of degree d. Additionally, assume that for every a ∈ U number of

−1 preimages πa (0), counted with multiplicities, is equal to d. Then the following identity is valid: [traceRω](a) = [tracef−aggω](0).

For the proof we will need the following lemma which is a precise quantitative form of the computation in the proof of Proposition 3.1.

Let V ⊂ C and U ⊂ C be bounded domains with boundaries, and f : V 7→ U a surjective holomorphic mapping defined in a neighborhood of the domain V . According to Rouche’s theorem, every regular value in the domain-image has the same number of preimages. Suppose that the domains V and U contain the origins, and the map f sending 0 ∈ U to 0 ∈ V has an isolated singularity at the origin in V . The latter means that 0 ∈ U is the only critical value and its preimage is the point 0 ∈ V only. Under these assumptions the mapping f is finite. Now, suppose that ω is a meromorphic form on V that is holomorpic on V \0.

Lemma 3.7. The function [tracef ω] is holomorphic on U\0 and its Laurent series at Chapter 3. Abel’s theorem 38

the origin is given by the following formula:

∞ ! 1 X k k−1 [tracef ω] = Res0(f ω)u , mult0f k=−N where u is a fixed coordinate on the domain-image, N is an integer, and mult0f the multiplicity of the mapping f at 0 ∈ V .

Proof of lemma. 1. The first part of the lemma is self-evident since the mapping f is a

finite covering over U\0, which implies that the function [tracef ω] is holomorphic on the complement U\0.

2. We first verify that Res0 tracef ω = mult0fRes0 ω. Indeed, let us choose coordinates z and w in the image and preimage so that near the origin 0 ∈ V the function f(z) = w =

n R R R z , where n = mult0f. Then Res0 tracef ω = tracef ω = tracef ω +...+ tracef ω, |w|=ε γ1 γn k−1 where γk (1 ≤ k ≤ n) is an arc on the circle |w| = ε between points ε exp(i n ) k R R and ε exp(i ). Notice that for any k, tracef ω = ω = Res0 ω. Therefore, n √ γk |z|= n ε n P R Res0 tracef ω = tracef ω = nRes0 ω. k=1γk ∞ P k 3. Now the kth term ak of the Laurent series ( aku ) of the function [tracef ω] is k=−N k−1 k−1 k−1 equal to Res0 u [tracef ω]du = Res0 u tracef ω = Res0 tracef f ω. Applying the formula from the previous step we complete the proof of the lemma.

Corollary 3.8. Assume that ω is a holomorphic form on V and mult0f ≥ ord0ω, where

ord0ω is the order of zero of ω at the origin. Then the function [tracef ω] is holomorphic

in the domain U and [tracef ω](0) = 0.

Corollary 3.9. Assume that ω has a pole at the origin and mult0f ≥ ord0ω, where

ord0ω is the order of pole of ω at the origin. Then the form tracef ω has at most a simple pole at 0 ∈ U.

Proof of Proposition 3.6. Let Af,g be the set of common zeroes of the functions f and

−1 g. Since for any point a in U the set Af,g ⊂ πa (0), the set Af,g is finite with at most d Chapter 3. Abel’s theorem 39

elements A1,...,An (n ≤ d). Also note that multAk f ≥ multAk g for any point Ak ∈ Af,g. S Let A be a finite set so that R : M 7→ U is a covering over U\A. For a ∈ U\(A R(Af,g)

−1 the zero level of the function f − ag is the union of two sets Af,g and R (a) =

T −1 {An+1(a),...,AN (a)} with the empty intersection Af,g R (a) = ∅. By definition, N [trace ω](a) = P ω (A (a)). Now ω (A (a)) = ω (A (a)). R d(R−a) k d(R−a) k d(f−ag) 1 +(f−ag)d 1 k k=n+1 g T ω gω The function f − ag vanishes at points Ak(a), so d(R−a) (Ak(a)) = d(f−ag) (Ak(a)) and N P gω [traceRω](a) = d(f−ag) (Ak(a)). k=n+1 We claim that the latter expression is equal to [tracef−aggω](0). Indeed, choose a neighborhood U ⊂ C of the origin and neighborhoods U1,...,Un of points A1,...,An,

respectively, such that the mapping (f −ag)|Uk : Uk 7→ U has an isolated singularity at the point Ak. For a generic a ∈ C, in the sense of Zariski topology, multAk (f −ag) = multAk g.

Corollary 3.8, yields to [trace(f−ag)| ω](0) = 0. Now, since the number of preimages Uk −1 πa (0) counted with multiplicities is equal to the degree of the mapping πa : Ma 7→ U, N n P gω P the trace function [tracef−aggω](0) = (Ak(a)) + [trace(f−ag)| ω](0). As d(f−ag) Uk k=n+1 k=1 we already observed the latter term is equal to 0 and this completes the proof.

Let R be a rational function on CP n and Γ˜ ⊂ CP n a projective (possibly singular) curve without multiple components. The next proposition shows that under a clearly necessary assumption R gives rise to a finite mapping Γ˜\Sing(Γ)˜ 7→ CP 1.

Proposition 3.10. The mapping R : Γ˜\Sing(Γ)˜ 7→ CP 1 is finite if and only if the function R is non-constant on each irreducible component of the curve Γ˜.

Proof. The necessity is self-evident, we now prove the sufficiency. Let R be a rational, ˜ non-constant on each component of the curve Γ function. We denote by Γ1,... Γr the ˜ ˜ irreducible components of Γ and by πν : Γν 7→ Γν the normalization mappings. There is a finite set A ⊂ Γ,˜ which contains the singular locus of the curve Γ,˜ such that the mapping ` ˜ π = (π1, . . . , πr) from the the disjoint union Γν to Γ is a holomorphic isomorphism Chapter 3. Abel’s theorem 40

˜ 1 outside of the set A. Since for each ν the mapping R ◦ πν : Γν 7→ CP is finite, we conclude that R : Γ˜ 7→ CP 1 is finite.

Corollary 3.11. A rational function R on an affine algebraic curve Γ ⊂ Cn without

multiple components gives rise to a finite mapping of Γ\Sing(Γ) to CP 1 if and only if R is non-constant on each irreducible component of Γ.

We call a rational function R on an affine algebraic curve Γ ⊂ C2 regular if it extends to a holomorphic, in the sense of algebraic varieties, mapping between the projective closure

Γ˜ of the curve Γ and CP 1. In terms of two relatively prime polynomials S and T in the S ˜ representation R = T , the function R is regular if no singular point of Γ belongs to both projective curves S = 0 and T = 0.

Define a residue form on an affine algebraic curve Γ ⊂ C2 without multiple components by the same formulas as in the case of a nonsingular curve Γ. The result is a holomorphic

1-form on the non-singular part of the curve. The next proposition shows that residue Qdx ∧ dy forms , where degQ ≤ degP − 3, on singular curves possess a vanishing trace dP property (see Proposition 3.1 ). Thus we will call them generalized holomorphic forms

and consider them as an analogue of holomorphic forms in the case of singular curves.

Proposition 3.12. Let Γ ⊂ C2 be an affine algebraic curve defined by a polynomial

P ∈ C[x, y] without multiple irreducible factors. Assume that R is a regular rational

1 Qdx∧dy function and the mapping R :Γ\Sing(Γ) 7→ CP is finite. Then traceR dP = 0 for degQ ≤ degP − 3.

Now we state an affine (plane) variant of Abel’s theorem to which we will prove a sort of

a converse statement. Let us call a couple of polynomials P1,P2 ∈ C[x, y] generic if two

2 affine curves P1 = 0 and P2 = 0 meet each other in degP1 × degP2 distinct points in C .

Proposition 3.13. Let Γ ⊂ C2 be an affine algebraic curve defined by a polynomial

P ∈ C[x, y] without multiple irreducible factors. Additionally assume that Γ transversely Chapter 3. Abel’s theorem 41

intersect the line at infinity. Let R be a polynomial such that the pair P,R is generic.

Qdx∧dy Then for any polynomial Q ∈ C[x, y] the trace form traceR dP is equal to F du with F ∈ C[u] a polynomial in a fixed coordinate u.

Remark 3.2. Later in the section 3.4, when we discuss a multidimensional Abel’s theorem,

we will remove the assumption of the transverse intersection of Γ with the line at infinity.

We will also weaken the genericity assumption on the pair of polynomials P,R.

S Proof of Proposition 3.12. Let R = T . In the view of Proposition 3.5 the statement is invariant under projective transformations. So we may assume that a) the projective closure Γ˜ of the curve Γ is transversal to the line at infinity; b) two pair of affine curves

P = 0 and T = 0, P = 0 and S = 0 meet each other in degP × degT and degP × degS points, counted with multiplicities, respectively; c) the function R attains finite, non-zero values at the points of Γ at infinity.

The proof consists of three steps. In Step 1, we prove the proposition assuming that the curve Γ is nonsingular. In Step 2, we assume that R is a polynomial and no smoothness assumption on Γ. In Step 3 we establish the proposition for R a regular rational function on the curve Γ, non-constant on each component of Γ.

Step 1. Under the above assumptions the curve Γ˜ ⊂ CP 2 is a compact one-dimensional complex manifold. As we previously noted the 1-form ω extends to a holomorphic 1-form ˜ on Γ. According to Abel’s theorem for curves, traceRω = 0. Step 2. Now assume that R = T is a polynomial in two variables. Choose a complex number a so that the curves P = 0 and R = a transversely meet each other at N =

2 degP ×degT points P1,...PN ∈ C at which both x and y could serve as local coordinates on Γ. Now, for sufficiently small ε and δ, the curve P = ε is non-singular and the curves R = a + δ and P = ε intersect each other transversely at precisely N points

P1(ε, δ),...,PN (ε, δ) with x(Pk) and y(Pk) holomorhic functions in variables ε and δ.

Qdx∧dy On the smooth curve P = ε consider the holomorphic form ωε = d(P −ε) , where degQ ≤

degP − 3. According to the Step 1, traceRωε = 0. Approaching ε to zero we find that Chapter 3. Abel’s theorem 42 traceRω is equal to zero near the point a. Since a ∈ C represents a generic, in the sense of Zariski topology, this completes the proof of the Step 2.

Step 3. By Corollary 3.11, the mapping R :Γ\Sing(Γ) 7→ CP 1 is finite and, therefore, the mapping R : R−1(C) 7→ C is finite as well. Since the rational function R = S/T gives rise to a finite mapping, for any a ∈ C, the polynomial S − aT is non-constant on each component of the curve Γ, and thus the mapping S − aT : R−1(C) 7→ C is finite. It is also not difficult to see that for the latter mappings the assumptions of Proposition 3.6 are valid (for a generic a ∈ C). We conclude that [traceRω](a) = [traceS−aT T ω](0) for a generic a ∈ C.

Now we shall show that traceS−aT T ω = 0 for any a ∈ C. By Proposition 3.13, the 1-

1 form traceS−aT T ω is polynomial on the complex line C ⊂ CP . The preimage of infinity (S − aT )−1(∞) consists of points at infinity of Γ.˜ Let A ∈ Γ˜ be a point at infinity, and v be a local coordinate at infinity of C ⊂ CP 1. Then near the point A the mapping 1 v = S−aT Γ˜. Since the function R attains finite, non-zero values at the infinity points of the curve Γ,˜ the polynomials S and T has equal orders n of pole at A, and thus the

1 function S−aT Γ˜ has a zero on order at least n at the point A. On the other hand, the the form T ω has a pole of order at most n. Applying the corollary 3.8 to all the points at infinity of the curve Γ we find that the form traceS−aT T ω has a pole of at most first order at infinity. However it is also a polynomial 1-form. Thus traceS−aT T ω = 0.

Proof of Proposition 3.13. We, essentially, repeat the arguments of steps 1 and 2 in the proof of the previous proposition.

Step 1. Assume that Γ ⊂ C2 ⊂ CP 2 is a smooth curve. Together with the transver- sality condition on the curve Γ˜ ⊂ CP 2 it follows that Γ˜ is a compact one-dimensional complex manifold. The form ω is a meromorphic 1-form on Γ˜ and the mapping R natu- rally extends to a finite type mapping R : Γ˜ 7→ CP 1. Notice that the it sends all the poles of the form ω into one point - infinity. Thus a meromorphic 1-form traceRω may have a Chapter 3. Abel’s theorem 43

singularity only at infinity. In other words, traceRω = F du with F ∈ C[u] a polynomial in a fixed coordinate u. We also observe that the degree of the polynomial F is bounded ˜ dx∧dy above by the maximum order of the pole of the 1-form ω on Γ. Since ω = Q dP and a dx∧dy ˜ 1-form dP is holomorphic on Γ, the degree of the polynomial F is bounded above by

the the maximum order of the pole of the function Q|Γ˜. However, the the sum of all its poles is equal to the sum of all its zeros, which is bounded above by the Bezout number

degF ≤ degQ × degP . It will be important in the Step 2.

Step 2. Choose a generic, in the sense of Zariski topology, a ∈ C such that the curves R = a and P = 0 transversely intersect each other at N = degP × degT points

2 P1,...PN ∈ C at which both x and y could serve as local coordinates on Γ = {P = 0}. Then for sufficiently small ε and δ, the curve P = ε is smooth and the curves R = a + δ and P = ε intersect each other transversely at precisely N points P1(ε, δ),...,PN (ε, δ) with x(Pk) and y(Pk) holomorhic functions in variables ε and δ. On the smooth curve

Qdx∧dy P = ε consider the 1-form ωε = d(P −ε) . According to Step 2, traceRωε = Fεdu, where

Fε ∈ C[u] is a polynomial of degree at most degR × degP . Approaching ε to zero, we

find that near the point a the function [traceRω] is equal to lim Fε. Since Fε ∈ C[u] ε→0

are polynomials of degrees uniformly bounded above, the function [traceRω] is also a polynomial (with the same bound on the degree).

Qdx∧dy It is instructive to examine a residue form τ = dP near a double point A of the curve Γ = {(x, y) ∈ C2|P (x, y) = 0}. By changing local coordinates we may assume that Qdx Qdx P (x, y) = xy. Then on the line y = 0 the form τ = 0 = and on the line x = 0 the Py x Qdy Qdy form τ = − 0 = − . Conclusion: if Q(A) 6= 0, then the residue form has a simple Px y pole on each branch passing through the double point A ∈ Γ and the corresponding

residues has the opposite signs; if Q(A) = 0, the residue form is holomorphic on each

branch of the curve Γ passing through the double point A. Chapter 3. Abel’s theorem 44

3.2 Application of Abel’s theorem: plane geometry

and Euler-Jacobi formula

Theorem 3.14 (Menelaus’ theorem). Let ABC be a triangle on the real plane R2 and l be

a line that intersect the sides AB, BC, CA of a triangle at points C1,A1,B1 respectively

(see the figure 3.1). Then BC1 CA1 AB1 = 1. CA A1B B1C

C_1 C B B_1 0 l(C)

A B_1 C

Figure 3.1: Menelaus’ theorem Figure 3.2: Integration path

Proof. We can think of a triangle ABC as of a singular cubic l1l2l3 = 0 on the complex

plane, where l1, l2, and l3 are linear polynomials with real coefficients that define lines AB, BC, and CA, respectively. The space of generalized holomorphic form on the cubic

is 1-dimensional and is generated by the form Ω = dx∧dy . We already know that the d(l1l2l3)

latter 1-form, when restricted to the lines l1 = 0, l2 = 0, and l3 = 0 has non-zero residues at the points A, B, and C. I also recall that the form Ω is holomorphic on each line

(including its infinity point) except for the points A, B, and C. Multiplying the form

Ω by a constant, we can choose a generalized holomorphic form ω in such a way that

ResAω|l1=0 = 1. Then ResAω|l2=0 = −ResAω|l1=0 − 1, ResBω|l1=0 = −ResAω|l1=0 = −1, etc. Let ` be a linear polynomial with real coefficients that defines the line l. Consider the mapping ` from the singular cubic l1l2l3 = 0 to the Riemann sphere. According to R R R ˜−1 the integral form of Abel’s theorem, ω + ω + ω = 0, where {γ1, γ2, γ3} = ` (γ) γ1 γ2 γ3 and γ is an oriented path from the origin to the infinity that does not pass through Chapter 3. Abel’s theorem 45

the points `(A), `(B), and `(C). For example, if the configuration is as shown on the

figure 3.1 we take γ as on the figure 3.2). Without loss of generality, we will assume that

γk ⊂ {lk = 0}. Now, the key point is that we can explicitly compute each integral. Indeed, if z is an

affine coordinate on the Riemann sphere defined by the equation l1 = 0, then ω|l1=0 =

dz − dz . Thus the first integral R ω is equal to ln( z(C1)−z(A) ) = ln( BC1 ). Summing z−z(A) z−z(B) z(C1)−z(B) C1A γ1 up with the other two integrals and exponentiating, we obtain that BC1 CA1 AB1 = 1. C1A A1B B1C

Theorem 3.15 (The Butterfly theorem). Let S and l be a circle and a line on the real

plane R2 which intersect each other at points A and B. Let M be a middle point of the segment AB; C and D are arbitrary points on the circle. Let E = CM T S, F =

DM T S, R = AB T EF , and T = CD T AB. Then MT = MR (see the figure 3.3).

D O E G

A TRM B CDAB

C F E F

Figure 3.3: Butterfly theorem Figure 3.4: The Klein model

Proof. Consider the circle S and the line l as a singular cubic Γ on the complex plane.

Again we choose the generalized holomorphic 1-form ω on the cubic Γ so that ω restricted

to the line l has the residue 1 at the point A ∈ l. Then ResBω|l = −ResAω|l = −1. Let π1

and π2 be two projections of the curve Γ from points D and E respectively to the line CF . M R R Application of the integral form of Abel’s theorem twice yields to ω|l + ω|S = 0 R arc CF T M T M R R R R R and ω|l + ω|S = 0. We conclude that ω|l = − ω|l = ω|l. The 1-form M arc CF R M T Chapter 3. Abel’s theorem 46

dz dz ω|l = z−z(A) − z−z(B) , where z is an affine coordinate. Integrating we find that for any B R1 two points A1,B1 ∈ l the integral ω|l = [A, B, A1,B1], where [A, B, A1,B1] is the A1 cross-ratio of the points A, B, A1,B1 on the line l. Finally, if A is a middle point of the segment AB, then from [A, B, R, M] = [A, B, T, M] it easily follows that RM = MT .

Remark 3.3. From the proof of the Butterfly theorem we see a recipe of how to solve the

following problem.

Given a segment CD and a point A (see the figure 3.4) on a line in the Klein model of

Lobachevsky geometry. Using a ruler only construct a segment AB that is equal, in the sense of Lobachevsky geometry, to the given one.

Indeed, take any point O on the circle S and construct the points OC T S = E and

OD T S = F . Then construct EA T S = G and GF T l = B, where l is the given line. Denote by X and Y the left and the right points, of intersection l T S, respectively.

Following the lines of the proof of the Butterfly theorem, we obtain that [X,Y,B,A] =

[X,Y,C,D]. The latter equality precisely means that the segments AB and CD have equal length in the Klein model of Lobachevskii geometry.

2 Theorem 3.16 (Euler-Jacobi formula). Suppose that Γ1, Γ2 ⊂ C are two affine curves defined by polynomials P1 and P2 ∈ C[x, y] of degrees m and n, respectively. Assume that the curves Γ1 and Γ2 meet each other transversely at degP1 × degP2 points A1,...,Amn.

Then for any polynomial Q ∈ C[x, y] of degree smaller or equal than m + n − 3 the following identity holds: Q Q (A1) + ... + (Amn) = 0, JP JP where JP is the jacobian of two functions P1 and P2.

Proof. We first consider the case when the curve P1 = 0 transversely intersects the line at infinity. Note that the statement of Euler-Jacobi formula is invariant under the affine Chapter 3. Abel’s theorem 47

change of coordinates. So without loss of generality we may assume that the curve P1 = 0

Qdx∧dy and coordinates lines have no common points at infinity. I claim that traceP = 0, 2 dP1 where Q ∈ C[x, y] is a polynomial of degree smaller or equal than m + n − 3. If a Qdx∧dy Qdx is a point at infinity of the curve P1 = 0, then the 1-form = 0 (considered dP1 (P1)y ˜ on the projective compactification Γ of the curve P1 = 0) has a pole of order at most

(m + n − 3) − (m − 1 − 2) = n. Since curves P1 = 0 and P2 = 0 have no common

points at infinity, the function P2 has a pole of order at least n at the point a. Thus ˜ 1 the multaP2 ≥ n, where P2 : Γ 7→ CP . Corollary 3.9 ensures that the trace form Qdx∧dy traceP has at most a simple pole at infinity. Proposition 3.13, however, asserts 2 dP1 that the latter is a polynomial 1-form. Thus Qdx∧dy = 0. dP1

Q Q Qdx∧dy I claim that [traceP ω](0) = (A1) + ... + (Amn), where ω = . Indeed, 2 JP JP dP1 mn Qdx∧dy P ω ω dP1 Qdx∧dy Q traceP [ω](0) = (Ak). Now (Ak) = = (Ak) = (Ak). 2 dP2 dP2 dP2 dP1∧dP2 JP k The general case can be reduced to the above by either slightly perturbing the coeffi-

cients of P1 or observing that, as follows from the transformation law in Proposition 3.20, the assertion of the Euler-Jacobi formula is invariant under projective transformations.

The latter approach we use in the proof of the multidimensional Euler-Jacobi formula in

Proposition 3.22.

For completeness we briefly indicate how to proceed with the first approach. Let

Q ∈ C[x, y] be a polynomial of degree m such that the curves P1 = 0 and Q = 0 have no

common points at the infinity line. Consider a polynomial Pa = P1 + aQ, where a ∈ C.

For a generic a, the curve Pa = 0 transversely intersects the line at infinity, and the curves

Pa = 0 and P2 = 0 transversely meet each other at mn points A1(a),...,Amn(a). The

functions x(Ak(a)) and y(Ak(a)) are analytic in a neighborhood U of the origin in C. For each a 6= 0, the Euler-Jacobi formula is valid for the curves defined by the polynomials

Pa and P2. By passing to the limit as a → 0, we confirm the validity of the Euler-Jacobi formula when a = 0. Chapter 3. Abel’s theorem 48

3.3 Abel’s theorem: continuation

So far we have discussed plane curves. In this section we will extend Propositions 3.12, 3.13

to the case of curves that are complete intersections in a multidimensional space. For

the precise statements see the Propositions 3.18, 3.19 below.

Let f1, . . . fk be holomorphic functions on an n-dimensional complex manifold U and ω be a holomorphic n-form.

Lemma 3.17. If the polyvector df1 ∧ ... ∧ dfk is not equal to zero at a point p ∈ U, then in a neighborhood Up of the point p there exists a holomorphic (n − k)-form τ such that τ ∧ df1 ∧ ... ∧ dfk = ω. The form τ is not unique, however its restriction to the T (n − k)-dimensional complex variety V Up with V = {f1 = f1(p), . . . , fk = fk(p)} is

T well-defined. Moreover, the restriction τ|V Up is uniquely determined by the validity of T τ ∧ df1 ∧ ... ∧ dfk = ω on the variety V Up.

The form τ|V we call a residue form and denote it by ResV ω. It is well-defined on the non-singular, (n − k)-dimensional part of the analytic set V ⊂ U. We will also use the

ω notation for V = {f1 = 0, . . . , fk = 0}. df1∧...∧dfk

Proof. By taking functions f1, . . . fk as the first k coordinates in a local coordinate system u1, . . . , un near the point P , we may assume that ω = gdu1 ∧...∧dun. In this coordinates P τ = gduk+1 ∧ ... ∧ dun + α, where α = cJ duJ is an arbitrary (n-k)-form and the J⊂{1,...,n} summation goes over all subsets J ⊂ {1, . . . , n} of n − k elements except for {k + 1, k +

2, k + 3, . . . , n} ⊂ {1, . . . , n}. The observation that α|V = 0 completes the proof.

n Example 1. Let x1, . . . , xn be a system of coordinates in C and ω = fdx1 ∧ ... ∧ xn. If k = n and p is a point of transversal intersection of hypersurfaces defined by the

f equations f1 = 0, . . . , fn = 0, then ResV ω = (P ), where Jf is the jacobian of the Jf

functions f1, . . . , fn. Chapter 3. Abel’s theorem 49

Let Γ ⊂ Cn be an affine algebraic curve defined by a system of polynomial equations

P1 = ... = Pn−1 = 0. We will assume that for almost all points P on the curve Γ, in the sense of Zariski topology, the rank of the Jacobian matrix Jac(P1,...,Pn−1)(P ) is maximal. Let Γ˜ ⊂ CP n be the projective closure of the curve. Qdx1 ∧ ... ∧ dxn Consider a residue form with a polynomial Q ∈ C[x1, . . . , xn] of dP1 ∧ ... ∧ dPn−1 degree at most degP1 + ... + degPn−1 − n − 1. If the curve Γ is smooth, the residue form extends to a holomorphic 1-form on the curve Γ˜ ⊂ CP n (e.g it follows from the transformation law in Proposition 3.5). Recall that a rational function R on the curve

Γ is called regular if it gives rise to a holomorphic, in the sense of algebraic varieties, mapping R :Γ\Sing(Γ) 7→ CP 1. Similarly to Proposition 3.12 the trace vanishing property remains valid for residue forms on a singular curve Γ. Proposition 3.13 could be readily generalized in this case as well. Below we state these generalizations and sketch their proofs.

Proposition 3.18. Suppose additionally that the projective variety in CP n defined by ˜ the homogenization of the polynomials P1,...,Pn−1 coincide with Γ. Let R be a regular

rational function, and assume that the mapping R :Γ\Sing(Γ) 7→ CP 1 is finite. Then the trace of Qdx1∧...∧dxn under the mapping R is identically zero for any polynomial Q ∈ dP1∧...∧dPk

C[x1, . . . , xn] of degQ ≤ degP1 + ... + degPk − n − 1.

An n-tuple of polynomials P1,...,Pn ∈ C[x1, . . . , xn] we call generic if n projective

n hypersurfaces, in the affine chart C with coordinates x1, . . . , xn, defined by the polyno- mials P1,...,Pn transversely meet each other in degP1 × ... × degPn points, all located in the affine chart Cn.

Proposition 3.19. Suppose, additionally, that Γ transversely intersect the line at infin- ity. Let R be a polynomial such that the n-tuple of polynomials P1,...,Pn−1,R is generic. Qdx1 ∧ ... ∧ dxn Then for any polynomial Q ∈ C[x1, . . . , xn] the trace form traceR is dP1 ∧ ... ∧ dPn−1 equal to F du where F ∈ C[u] is a polynomial in a fixed coordinate u. Chapter 3. Abel’s theorem 50

Sketch of proof of proposition 3.19. We follow the lines of the proof of the proposition 3.13.

Step 1. If the curve Γ ⊂ Cn is smooth, then the genericity assumption guarantees

that the curve Γ˜ ⊂ CP n is a compact one-dimensional complex manifold. The mapping R : Γ˜ 7→ P 1 sends all the poles of the meromorphic 1-form ω = Qdx1∧...∧dxn to infinity. C dP1∧...∧dPn−1

Thus traceRω = F du. The degree of the polynomial F is bounded by maximum order of the pole of the function Q on the curve Γ.˜ However, the sum of all the poles is equal

to the sum of all the zeroes of the function Q, which is bounded by the Bezout number

degP1 × ... × degPn−1 × degQ.

Step 2. If the curve Γ is not smooth, then one considers curves Γε¯ that are given

by the equations P1 = ε1,...,Pn−1 = εn−1. For a generic, sufficiently small vector

ε¯ = (ε1, . . . , εn−1) the curve Γε¯ is nonsingular and the n-tuple of polynomials P1 −

ε1,...,Pn−1 − εn−1 is generic. Applying the first step to the curve Γε¯, approachingε ¯

to zero, and observing that degFε¯ ≤ N, where N is independent fromε ¯, we obtain the result.

Sketsh of Proof of Proposition 3.18. With the transformation law in Proposition 3.20 below the proof goes along the same lines as the proof of Proposition 3.12

Now we examine the behavior of residue forms with respect to projective transforma-

tions. We will prove it in the following setup.

Let Γ ⊂ Cn be an affine algebraic k-dimensional variety defined by a system of

polynomial equations P1 = ... = Pn−k = 0. We will assume that for almost all points P of the curve Γ, in the sense of Zariski topology, the rank of the Jacobian matrix

n Jac(P1,...,Pn−k)(P ) is maximal. Now, a projective transformation of CP is an element

n n L of P GLn+1(C). In a fixed affine chart C ⊂ CP with coordinates (x1, . . . , xn) the

n l1 ln image of a point p ∈ C is L(p) = ( l (p)),..., l (p)), where l, l1, . . . , ln ∈ C[x1, . . . , xn] are polynomials of the first degree, whose coefficients are the rows of the matrix L. Let Chapter 3. Abel’s theorem 51

˜ degPj l1 ln degPj ∗ ˜ degQ l1 ln degQ ∗ Pj = l Pj( l ,..., l ) = l L (Pj) and Q = l Q( l ,..., l ) = l L Q. Under these notations

Proposition 3.20. The following formula is valid  Qdx ∧ ... ∧ dx  lN−degQQ˜dx ∧ ... ∧ dx L∗ 1 n = 1 n , ˜ ˜ dP1 ∧ ... ∧ dPn−k dP1 ∧ ... ∧ dPn−k

where N = degP1 + ... + degPk − n − 1.

For the proof we will need the lemma below

Lemma 3.21. For any n+1 linear polynomials l1, . . . , ln+1 ∈ C[x1, . . . , xn], the following identity is valid: n+1 X k (−1) l1dl2 ∧ ... ∧ dlk−1 ∧ dlk+1 ∧ ... ∧ dln+1 = det A dx1 ∧ ... ∧ dxn+1, k=1

where coefficients of polynomials l1, . . . , ln+1 respectively form the first n + 1 rows of the matrix A.

Qdx1∧...∧dxn Proof of Proposition 3.20. By definition = τ so that τ ∧ dP1 ∧ ... ∧ dPn−k = dP1∧...∧dPn−k

Qdx1 ∧ ... ∧ dxn. For computational convenience we rewrite the right hand-side as

dx1 dxn ∗ x1 . . . xnQ ∧ ... ∧ , and apply the operator L to both sides. We obtain: x1 xn

∗ ∗ ∗ ∗ ∗ ∗ L (τ) ∧ L (dP1) ∧ ... ∧ L (dPn−k) = L (x1 . . . xnQ)L (d ln x1) ∧ ... ∧ L (d ln xn).

˜ ∗ Pj dP˜ ∗ dlj dl Now L (dPj)|L−1Γ = d degP |L−1Γ = degP , and L (d ln xj) = d ln lj − d ln l = − , and l j l j lj l L∗Q = ldegQQ˜. Substituting these and applying the identity from the lemma above to

∗ ∗ −1 compute L (d ln x1) ∧ ... ∧ L (d ln xn), we find that on the variety L Γ the following is

∗ ˜ ˜ degP1+...+degPk−n−1 ˜ ˜ ˜ valid L (τ) ∧ dP1 ∧ ... ∧ dPn−k = l QdP1 ∧ ... ∧ dPn−k This completes the proof.

Proof of the lemma 3.21. We can think of coefficients of l1, . . . , ln+1 as vectors of an n+1- dimensional vector space V . Note that n+1 X k (−1) l1dl2 ∧...∧dlk−1 ∧dlk+1 ∧...∧dln+1 = (a0 +a1x1 +...+anxn) dx1 ∧...∧dxn+1, k=1 Chapter 3. Abel’s theorem 52

where coefficients ak are skew-symmetric polylinear functions on V . Thus, up to multi- plication by a constant, they all coincide with the det A. To determine the constants, we

compute the right hand side of the above identity for l1 = x1, . . . , ln = xn, ln+1 = 1. We

conclude that a1 = ... = an = 0 and a0 = det A.

Proposition 3.22 (Euler-Jacobi formula). Suppose that P1,...,Pn ∈ C[x1, . . . , xn] is a generic n-tuple of polynomials and the affine hypersurfaces defined by the polynomials

P1,...,Pn transversely meet each other at points A1,...,AN , where N is the Bezout

number degP1 × ... × degPn. Then for any polynomial Q ∈ C[x1, . . . , xn] of degree at

most degP1 + ... + degPn − n − 1 the following identity holds:

Q Q (A1) + ... + (AN ) = 0, JP JP where JP is a jacobian of the functions P1,...,Pn.

Proof. As follows from the genericity assumptions, the projective hypersurfaces defined

by the polynomials P1,...,Pn−1 transversely meet each other at a generic, in Zariski

sense, point of a projective curve Γ˜ ⊂ CP n. Let Γ ⊂ Cn be the affine part of Γ˜ in

the affine chart with coordinates x1, . . . , xn. By the example 1 and Proposition 3.20, the assertion of the Euler-Jacobi formula is invariant under projective transformations of

CP n. So without loss of generality we may assume that Γ transversely meets the infinity

1 hyperplane. Consider the mapping Pn :Γ 7→ CP . I claim that tracePn ω = 0, where ω = Qdx1∧...∧dxn . Indeed, it follows from the transformation formula 3.20, applied for a dP1∧...∧dPn−1 projective transformation L with a generic l, that at the infinity points of Γ the form ω

has a pole of order at most N − degQ = degP1 + ... + degPn−1 − n − 1 − degQ ≤ degPn.

At the same time the function Pn has a zero of order at least degP at the infinity points

1 of Γ. Now by Proposition 3.13 the form tracePn ω is polynomial on C ⊂ CP , and by the corollary 3.9 it has at most a simple pole at infinity. Thus tracePn ω = 0. Notice that Chapter 3. Abel’s theorem 53

N P ω ω Qdx1∧...∧dxn [traceP ω](0) = (Ak), and = . From the example 1 we conclude n dPn dPn dP1∧...∧dPn k=1 Q Q that (A1) + ... + (AN ) = 0 JP JP

3.4 Multidimensional Abel’s theorem and general-

ized holomorphic forms

We will start with a definition. A holomorphic mapping F : M 7→ N between two

analytic varieties of the same dimension is called finite if 1) every point of the image N

has a finite number of preimages, 2) there is an analytic hypersurface A ⊂ N, such that

−1 the mapping F : M\F (A) 7→ N\A is a finite covering of complex manifolds. If Ua is

−1 a small neighborhood of a point a ∈ N\A, then f (Ua) is a disjoint union of connected

sets U1,...,Ud, for some d ∈ N, such that the mapping f : Uk 7→ U is a biholomorphic

−1 isomorphism. Denote the inverse f |U by gk. Given a holomorphic n-form ω on the

smooth part of the variety M we define a trace form traceF ω on A, which is a holomorphic

n-form on the complex manifold N\A. In a neighborhood Ua it is given by the following formula d X ∗ tracef ω = gkω. k=1 A holomorphic mapping F : M 7→ N that satisfies only the first condition we call

weakly finite.

Proposition 3.23. If N is a smooth complex manifold then a proper, weakly finite map-

ping F : M 7→ N is finite.

For the proof we will use the following fact from the complex analysis

(Remmert’s proper mapping theorem) A proper mapping between analytic varieties sends analytic subvarieties onto analytic subvarieties. Chapter 3. Abel’s theorem 54

Remark 3.4. In our applications we will need only Remmert’s proper mapping theorem

for algebraic mappings between algebraic varieties, and for the proper, weakly finite

mappings between smooth analytic varieties.

Proof of Proposition 3.23. Without loss of generality we may assume that N is con-

nected. Let Sing(M) be the singular locus of the variety M and V ⊂ M be the hy- persurface of critical points of the mapping F |M\Sing(M). By Remmert’s proper mapping theorem A = F (Sing(M) S V ) ⊂ N is an analytic subvariety, which should be of codi- mension 1 (since F is a weakly finite mapping). Now F : M\F −1(A) 7→ N\A is smooth proper mapping between two smooth real manifolds with a connected image, therefore a mapping degree is well-defined. As usual in complex analysis all the sings in the defi- nition of the mapping degree are ” + ” so every point in N\A has the same number of preimages.

Theorem 3.24. [Multidimensional Abel’s theorem 1] Given a proper, finite, holomorphic

mapping f : M 7→ N between two complex n-dimensional manifolds and a holomorphic

n-form ω on the manifold M. Then there is a holomorphic continuation of the trace form

tracef ω to the whole manifold N.

n ∗ n Let I ×I ⊂ C ×C be a punctured polycylinder |x1| ≤ ε1,..., |xn| ≤ εn, 0 < |y| ≤ ε,

n where x1, . . . , xn and y are coordinates in C and in C respectively, and ε1, . . . , εn, ε are

some positive numbers. Denote by dµ the volume form dx1 ∧dx1 ∧... dxn ∧dxn ∧dy ∧dy

on Cn+1 = Cn × C. We will need the following

Lemma 3.25. Let f be a holomorphic function on the punctured polycylinder that satis-

fies the L2 growth condition R |f|2dµ < ∞. Then there is a holomorphic continuation In×I∗ of the function f to the whole polycylinder In × I.

Proof of Theorem 3.24. In the complement to the hypersurface of critical values A the trace form traceF ω is holomorphic and, locally, tracef ω = (g1 + ... + gd)dz1 ∧ ... ∧ dzn, Chapter 3. Abel’s theorem 55

where z1, . . . , zn are coordinates in a neighborhood of a non-critical value of the mapping

2 2 2 2 F . Using the estimate |g1 + ... + gd| ≤ (|g1| + ... + |gd|) ≤ n(|g1| + ... + |gd| ) (the first is the triangle inequality, the second is Cauchy-Schwarz inequality), we see that

2 the function g1 + ... + gd satisfies the L -growth condition near a smooth point of the hypersurface A. Thus the trace form admits holomorphic continuation to the smooth locus of A. The application of Hartog’s theorem completes the proof.

Proof of the lemma 3.25. The lemma follows from a straightforward computation. We ∞ P n expand f in Laurent series f(x1, . . . , xn, y) = f(x, y) = cn(x)y , where cn are holo- −∞ morphic functions on In. To compute R |f|2d arg(y) we use the identities |y|=r

1 Z ymy¯ndarg(y) = r2nδn , 2πr m |y|=r

n where r is a positive real number, n and m are integers, and δm is the Kronecker symbol. ∞ R 2 P 2 2n 2 It gives that |f| d arg(y) = 2πr( |cn| r ). Thus the L -growths condition implies |y|=r −∞ that coefficients cn with negative n vanish.

Let f : U 7→ Cn be a holomorphic mapping defined in a neighborhood of a bounded

domain U ⊂ Cn with a boundary. Assume that the image of the mapping f is a domain

V ⊂ Cn with a boundary. According to the multidimensional Rouche’s theorem, the mapping f : U 7→ V is weakly finite. It is also proper since f maps the boundary ∂U

into the boundary ∂V . We conclude that the following corollary is valid

Corollary 3.26. For ω a holomorphic n-form on U, the trace form tracef ω admits holomorphic continuation to the whole domain V .

n Let f = (f1, . . . , fn): U 7→ C be as above. Suppose that the analytic set Γ =

{f1 = ... = fk = 0} is a complete intersection. The image Imf|Γ belong to a certain

n coordinate k-dimensional subspace Vk ⊂ C . We will assume that Imf|Γ is not a subset of the hypersurface A, where A is an ”exceptional” hypersurface from the definition of Chapter 3. Abel’s theorem 56

n the finite mapping f : U 7→ C . Thereby the mapping g = f|Γ is also finite. That brings us to

Proposition 3.27 (Multidimensional Abel’s theorem 2). For any holomorphic n-form

ω on the domain U there is a holomorphic continuation of the trace form tracegResΓω T to Vk V .

Proof of Proposition 3.27. The coordinate-free definition of residue forms ensures that

for any complete intersection D = {f1 = ... = fk = 0} in a complex n-dimensional manifold M, for any holomorphic n-form ω on M, and a biholomorphic isomorphism

π : N 7→ M between complex manifolds the following identity is valid:

−1 ∗ ∗ (π ) Resπ∗Dπ ω = ResDω, (3.2)

∗ ∗ ∗ where π D ⊂ N is a complete intersection π f1 = ... = π fk = 0.

We now return to the proof of Proposition. Let y1, . . . , yn be coordinates in the image,

in which f has the form y1 = f1, . . . , yk = fk. The k-dimension plane Vk is the complete d P ∗ intersection y1 = ... = yk = 0. Locally tracef ω = gkω, and the term-wise application k=1

of the identity (3.2) implies that tracegResΓω = ResVk tracef ω. By the proposition 3.26

the trace form tracef ω is holomorphic on V thereby ResVk tracef ω is holomorphic on T Vk V .

We are ready to state our final form of Abel’s theorem. Let Γ ⊂ M be a locally

complete intersection in a complex manifold M. I remind that an analytic subvariety

Γ in a complex n-dimensional manifold M is called a locally complete intersection if,

in a sufficiently small n-dimensional neighborhood of any its point, Γ is a transversal

intersection of k analytic hypersurfaces D1,...,Dk, where k is the codimension of Γ in M.

Suppose that ω is a holomorphic form of the highest degree on the non-singular part of Γ, and, in a sufficiently small n-dimensional neighborhood U of a point in Γ, the form Chapter 3. Abel’s theorem 57

T T ω = ResD1 ... Dk Ω for Ω a holomorphic form on U. It is not difficult to check that the above property of the form ω does not depend on the choice of the local representation of T T Γ as a transversal intersection D1 ... Dk. Under these notation the following theorem is valid

Theorem 3.28 (Multidimensional Abel’s theorem 3). For any proper, finite, holomor-

phic mapping F :Γ 7→ N onto a connected complex manifold N, the trace form traceF ω admits holomorphic continuation to the whole N.

Recall a mapping between topological spaces is called open, if the image of an open set is open. We will need the following

Lemma 3.29. Any holomorphic, weakly finite mapping F :Γ 7→ N is open.

Proof of the lemma 3.29. This is a local question. We will show that the restriction of

F to a sufficiently small neighborhood of a prescribed point in Γ is an open mapping.

Let U be an n-dimensional neighborhood of a point P in Γ with the following properties:

2 1) it is a disk ||z|| ≤ ε in a coordinate chart z1, . . . , zn centered at P ; 2) the variety T Γ U is a complete intersection defined by k holomorphic in U functions f1, . . . , fk; 3)

the mapping F is given by n − k holomorphic in U functions fk+1, . . . , fn; 4) the value F (P ) is isolated, that is F −1(F (P )) T U = P .

˜ n It follows that the mapping F = (f1, . . . , fn): U 7→ C has an isolated singularity at the origin (recall that in the chosen coordinate chart the point P corresponds the origin).

By choosing a sufficiently small disk V around F (P ) and redefining U to be U T F˜−1(V ) we obtain an open, surjective mapping F˜ : U 7→ V . Thus its restriction to Γ T U is also open.

Proof of Theorem 3.28. Since F is proper, finite, and open, it induces a branched cov-

ering F :Γ 7→ N of degree d in the sense of page 18. Now take any point P in N. Chapter 3. Abel’s theorem 58

−1 Its preimage F (P ) is a collection of points P1,...,Pν together with multiplicities d1, . . . , dν, which sum is d. As in the end of the lemma 3.29, we choose a sufficiently small disk V in a coordinate chart of the point P with P ∈ V , open n-dimensional sets

U1,...,Uν ⊂ D in coordinate charts of points P1,...,Pν respectively with Pk ∈ Uk, so that F (Uk) = V and the boundary of Uk is mapped into the boundary of the disk V . Now near the point P the trace trace ω = trace ω + ... + trace ω. The term-wise F FU1 FUν application of Proposition 3.27 completes the proof.

In our applications forms on complete intersections, as in Proposition 3.28, occur as

residues of globally defined meromorphic forms of the highest degree on complex man-

ifolds. These residues are coordinate-free analogues of the forms considered in the sec-

tions 3.1, 3.3.

Let M be a complex manifold and ω be a meromorphic form of the highest degree on

M. Suppose that D + W is the polar divisor of the form ω, where D ⊂ M is an analytic

hypersurface. In particular, it means that the form ω has a simple pole along the hyper- surface D ⊂ M. Also notice that D T W is a codimension one analytic subvariety of D.

One can define a meromorphic form ResDω of the highest degree on the analytic subvari- T ety D as follows. In a small neighborhood Ua of a smooth point a ∈ D\Sing(D)\(D W ) the divisor D = {f = 0}, where f ∈ O(Ua) is a holomorphic function and df 6= 0 at the

ω˜ n,0 point a; the form ω = f whereω ˜ ∈ Ω (Ua) is a holomorphic n-form. Take ResDω to ω˜ be df . It is not difficult to check that the form ResDω does not depend on the choice of

the function f. The form ResDω is called a residue form. In local coordinates x1, . . . , xn

gdx1∧...∧dxn k gdx1∧ xk−1∧dxk+1...∧dxn the form ω = f and the residue form ResDω = (−1) f 0 at a xk

point of the divisor D = {f = 0} where x1, . . . , xk−1, xk+1, . . . , xn form local coordinates.

If D1 + D2 + W is the polar divisor of the form ω, where D1,D2 ⊂ M are analytic hy- persurfaces, then one may repeat the construction (under some necessarily assumptions)

with the meromorphic form ResD1 ω on the smooth part of the divisor D1. The next proposition shows that this invariant construction locally coincide with the construction Chapter 3. Abel’s theorem 59

of residue forms in the section 3.3.

Proposition 3.30. Given a meromorphic form ω with a simple pole along the divisors

D1,...,Dk and with the polar set D1 + ... + Dk + W . Assume that at a generic point T T of the analytic variety V = D1 ... Dk the divisors D1,...,Dk transversely meet each

ω˜ other. Then locally ResD (... ResD ((ResD ω))) = , where fj = 0 are defining k 2 1 dfk∧...∧df1 ω˜ equations equation for the divisors Dj, and ω = . f1...fk

Proof. The proof is a straightforward computation in a suitable coordinate system near a generic point of V . We take f1, . . . , fk as the first k coordinate functions.

The lemma implies that the procedure of taking the residue in ”skew-symmetric” in

the following sense ResD2 ((ResD1 ω) = −ResD1 ((ResD2 ω). Whenever a particular sign of

T T the residue form is not important for us we will denote the residue by ResD1 ... Dk ω. We now state multidimensional analogues of Propositions 3.18, 3.19 from the sec-

tion 3.3. Corollary 3.31 is the analogue of Proposition 3.18 and is a direct consequence of

Theorem 3.28, the transformation law in Proposition 3.20, and the absence of holomor-

phic forms of the highest degree on the projective space. Corollary 3.32 is the analogue

of Proposition 3.19 and is a consequence of Theorem 3.28. Below by a generic point we

mean a generic point in Zariski sense.

n Corollary 3.31. Suppose that D1,...,Dk are hypersurfaces in CP that meet each other T T transversely at a generic point of the projective subvariety Γ = D1 ... Dk. Assume

n n that in the affine chart C ⊂ CP with coordinates x1, . . . , xn hypersurfaces D1,...,Dk

are defined by polynomials P1,...,Pk. Then for any proper, finite, rational mapping

n−k R :Γ 7→ CP and a polynomial Q of degQ ≤ degP1 + ... + degPk − n − 1 the following

Qdx1∧...∧dxn identity is valid: traceR = 0. dP1∧...∧dPk

n Corollary 3.32. Suppose that D1,...,Dk are hypersurfaces in C that meet each other T T transversely at a generic point of the affine subvariety Γ = D1 ... Dk. Assume that Chapter 3. Abel’s theorem 60

in affine coordinates x1, . . . , xn the hypersurfaces D1,...,Dk are defined by polynomials

n−k P1,...,Pk, and R :Γ 7→ C is a proper, finite, polynomial mapping. Then for any poly-

Qdx1∧...∧dxn nomial Q the trace form traceR is F du1 ∧...∧dun−k with F ∈ [u1, . . . , un−k] dP1∧...∧dPk C n−k a polynomial depending on Q, in a fixed affine coordinates u1, . . . , un−k in C .

Proof of Corollary 3.32. By Theorem 3.28, the trace of Qdx1∧...∧dxn under the mapping dP1∧...∧dPk n−k R is F du1 ∧ ... ∧ dun−k with F holomorphic on C . We now show that F is, in fact, a polynomial. Denote the hypersurface of critical

n−k values by Σ ⊂ C . Let `1, . . . , `n−k−1 be polynomials of the first degree which zero locus define a line l with l * Σ. The preimage R−1(l) ⊂ Γ is a (singular) affine curve

n ω Γl in . On the curve consider meromorphic 1-form ωl = ∗ ∗ , where C d(R `1)∧...∧d(R `n−k−1)

Qdx1∧...∧dxn ω = . The 1-form ωl is meromorphic on the normalization of the projective dP1∧...∧dPk closure of Γl, and thus, by Abel’s theorem for curves, the trace of ωl under mapping R is

meromorphic on the line l. We conclude that traceRωl is a polynomial. It follows from

F du1∧...∧dun−k the identity (3.2) on the page 56 that traceRωl = , and we conclude that d`1∧...∧d`n−k−1 the function F restricted to the line l is a polynomial.

By Proposition 6.2 applied to both Re F and Im F , the degrees of all such polynomials

are bounded above by a single constant, and thus F is a polynomial (see page 107 for

the statement and Proposition 2.21 for a more general statement).

In the view of Propositions 3.4 and Corollaries 3.31 3.32 we define certain generalized

holomorphic forms of the top degree on (singular) k-dimensional affine varieties without

multiple components.

Let Γ ⊂ Cn be a k-dimensional affine algebraic variety without multiple components and ω a k-form holomorphic on the non-singular locus of the variety Γ. We call ω a generalized holomorphic if for any proper finite polynomial mapping f :Γ 7→ Ck the

trace form tracef ω = Pf du1 ∧ ... ∧ dun−k with Pf ∈ C[u1, . . . , un−k] a polynomial in a

n−k fixed affine coordinates u1, . . . , un−k in C . Chapter 3. Abel’s theorem 61

The next proposition is almost self-evident.

Proposition 3.33. Let Γ ⊂ Cn be a k-dimensional affine algebraic variety without mul- tiple components and ω a k-form holomorphic outside of a hypersurface that contains

the non-singular locus of the variety Γ. Then ω is a generalized holomorphic form

if and only if for any proper finite polynomial mapping f :Γ 7→ Ck the trace form

tracef ω = Pf du1 ∧ ... ∧ dun−k with Pf ∈ C[u1, . . . , un−k] a polynomial in a fixed affine

n−k coordinates u1, . . . , un−k in C .

Proof. Suppose that ω is holomorphic on Γ\Σ, where Σ is a hypersurface that contains

the singular locus of Γ. We need to show that, if the trace of ω is under any proper finite

polynomial mapping is polynomial, then ω is holomorphic at points of Σ where Γ is non-

singular. This is done by considering a projection along a generic (n − k)-dimensional

plane onto some Ck.

Let Γ ⊂ CP n be a k-dimensional projective algebraic variety without multiple com- ponents and ω a k-form holomorphic on the non-singular locus of the variety Γ. We call

ω a generalized holomorphic form if, given an affine chart Cn ⊂ CP n, the restriction of T n ω to Γ C is a generalized holomorphic form. It is then a theorem that

Given a projective k-dimensional variety Γ ⊂ CP n. Then a k-form ω holomorphic on the non-singular part of Γ is a generalized holomorphic if and only if for any proper, finite,

k rational mapping R :Γ 7→ CP the trace traceRω = 0. Chapter 4

Differentiation rule

Unless otherwise stated, the manifolds, mappings, forms, vector fields, and functions

considered below are assumed sufficiently smooth. All the results below have complex-

analytic analogues where the smooth data is replaced by the analytic one. Proofs, essen-

tially, remain unchanged.

4.1 One-dimensional case

Let γ ⊂ R2 = R × R be a graph of a real-valued function g defined on an interval of

the real line, and U a connected domain in the space of all lines on R2. Suppose that U belong to the affine chart in which every line has an equation x = ay + b, where (x, y)

are fixed coordinates in R2 = R × R and a, b are some real numbers. We will think of (a, b) as coordinates on the domain U.

Suppose that the map F : U 7→ γ sending a point in U, which is a line ` on the plane, into the intersection ` T γ is well-defined, surjective, and of the maximal rank everywhere. Let ω and f be a 1-form and a function on γ, respectively. Let ρ be the

∗ function on U determined from the identity F ω|a=a0 = ρ(a0, b)db. Denote the functions F ∗f and F ∗g by f˜ andg ˜, respectively.

62 Chapter 4. Differentiation rule 63

Proposition 4.1 (Differentiation rule 1). The following identities are valid

∂ ∂ ρ = (˜gρ) (4.1) ∂a ∂b

∂ ∂ f˜ =g ˜ f˜ (4.2) ∂a ∂b

Proof. For the proof we will use the necessity condition from the lemma below. According to implicit mapping theorem for a fixed point p ∈ γ the preimage F −1(p) is a smooth

curve in U. Vectors tangent to these curves form the kernel of F∗. The equation of the curve F −1(p) is x(p) = ay(p) + b, which is a line on the plane with coordinates (a, b).

The vectors tangent to these lines are given by the following formula (−1, y(a, b)), where

y(a, b) = F ∗y = F ∗g.

Now let F ∗ω = ρ(a, b)da + h(a, b)db. Then the first condition in the necessity of

∂h(a,b) Lemma 4.2 translates into h(a, b) = y(a, b)g(a, b), and the second translates into ∂a = ∂g(a,b) ∂b . That proves the first identity. ˜ The second identity is the direct corollary of ivdf = 0 with v the vector field

(−1, y(a, b)) belonging to the kernel of F∗.

Remark 4.1. Naturally identifying the real line with coordinate b and the real line with

coordinate x, we interpret the form ρ(a0, b)db as the trace of ω under the parallel projec- tion along the line x − a0y = 0.

Lemma 4.2. Assume that π : M 7→ N is a surjective mapping of the maximal rank everywhere between two connected real manifolds, and α is a differential form on M.

Then α = (π−1)∗β if and only if the following two conditions hold:

1. For any vector v in M such that π∗v = 0 ivα = 0;

2. For any vector v in M such that π∗v = 0 iv(dα) = 0. Chapter 4. Differentiation rule 64

Proof. Necessity. Assume that α = (π−1)∗β for a smooth form β on N. If v is a vector

∗ ∗ such that π∗v = 0, then ivα = π (iπ∗v)β = 0 and iv(dα) = π (iπ∗vdβ) = 0. Vector v with the property π∗v = 0 we will call vertical. At each point those are just the kernel of π∗. Sufficiency. Assume that α is a smooth form on M satisfying 1) and 2). Notice that the dimension of M is greater than the dimension of N because π is surjective. Let dimM = n+k, dimN = n, k > 0. We want to prove the existence of a smooth form β on

N such that α = (π−1)∗β. This is a local question, so we can restrict ourselves by a small neighborhood of a point p in M. A straightforward computation in local coordinates completes the proof. See [8] for details.

4.2 Multidimensional case

Given a function g : M 7→ R and a vector field v on a real manifold M, let ht denote the phase flow corresponding to v. Consider the map hgt sending a point p ∈ M to a point

hg(p)t(p).

Proposition 4.3. The map from M × R to M × R sending a point (p, t) to (hgt, t) has

the maximal rank at the points of the zero section M × {0} ⊂ M × R.

Proof. Fix a point p ∈ M. In coordinates x1, . . . , xn, t near the point (p, 0) in which the

∂ gt vector field v is the coordinate one the mapping h sends a point (x1, . . . , xn, t) to ∂x1

the point (x1 + tg, . . . , xn). Thus the mapping in question sends a point (x1, . . . , xn, t) to

the point (x1 + tg, . . . , xn, t). Its jacobi matrix J at the point (p, 0) is an (n + 1) × (n + 1) matrix that differs from the identity only by the last element of the first row. We conclude

that the rank of J is maximal.

The next corollary immediately follows from the proposition above and the inverse func-

tion theorem. Chapter 4. Differentiation rule 65

Corollary 4.4. Let p ∈ M be a point. Then there are connected open sets Ut ⊂ M

gt parameterized by a sufficiently small t ∈ R so that h : U0 7→ Ut is a diffeomorphism and S p ∈ U0. The union V = (Ut, t) is an open connected subset in M × R, which can be chosen arbitrary small.

The next Corollary we will need in the proof of the first main theorem. It is a version of

Corollary 4.4 when the vector field v depends on some space of parameters Rm, and that is why we present it here. In this case, the mapping hgt depends on v ∈ Rm as well, and

gt we denote it by hv .

v Corollary 4.5. Let p ∈ M be a point. Then there are connected open sets Ut ⊂ M

m v parameterized by a sufficiently small t ∈ R and v ∈ R so that U0 = U0 is a single

gt v neighborhood of the point p and hv : U0 7→ Ut is a diffeomorphism. The union V = S v m (Ut , v, t) is an open connected subset in M × R × R, which can be chosen arbitrary small.

m Proof of Corollary 4.5. Let y = (y1, . . . , yn) be a fixed system of coordinates in R .

On M × Rm we consider the vector field w = (v, 0) and the function G = π∗g, where

π : M × Rm 7→ M is the projection. Apply Corollary 4.4 at the point (p, 0). We

Gt find (arbitrary small) connected (n + m)-dimensional domains Vt so that h maps V0

Gt diffeomorphically onto Vt, where the mapping h is constructed for the vector field w on

M × Rm. Note that hGt preserves the horizontal sections of M × Rm and coincide with

gt hv on M × {v}.

0 0 To complete the proof, we choose a connected subdomain V0 ⊂ V0 of the type U0 ×U0

0 m 0 m gt with U0 ⊂ M and U0 ⊂ R . For v ∈ U0 ⊂ R the mapping hv sends U0 diffeomorphically onto its image.

Let ω be an arbitrary form of the highest degree on M. The following rule is valid. Chapter 4. Differentiation rule 66

Proposition 4.6 (Differentiation rule). For a sufficiently small neighborhood U of any

d gt −1 prescribed point, there is a sufficiently small time interval in which dt [ω((h )∗ ξ)] = gt −1 ∗ −[Lv((h ) ) (gω)](ξ), where ξ is a multivector at a point P of U.

Corollary 4.7. It is true that ds [ω((hgt)−1ξ)]| = (−1)s[Ls(gsω)](ξ), dts ∗ t=0 v where s ∈ N.

d2 gt −1 d gt −1 ∗ Proof of the corollary 4.7. Indeed, dt2 [ω((h )∗ ξ)] = [ dt (−Lv((h ) ) (gω))](ξ) = d gt −1 ∗ 2 gt −1 ∗ −[Lv dt ((h ) ) (gω))](ξ) = [Lv((h ) ) (gω)](ξ). Substituting t = 0 we obtain the formula for s = 2. For general s we prove by the induction.

For the proof of the differentiation rule we will need a certain Liouville-type formula.

Given a vector field on a real manifold M and a k-form α. Let D be a diffeomorphic

image in M of a k-dimensional disk. Denote by Dt the image of D under the phase flow of the vector field v and by V (t) the integral R α. Under these assumptions Dt d Z V (t) = L α, (4.3) dt v Dt

where Lv is the Lie derivative.

Remark 4.2. If M is a domain in the euclidian space with the standard volume form α =

dx1 ∧ ... ∧ dxn, then Cartan’s identity implies that Lvα = (div + ivd)α = divα = div(v)

d R and the formula (4.3) transforms into a well-known Liouville formula dt V (t) = div(v)α. Dt

Proof of the proposition 4.6. Fix a point in M and take domains U = U0 and V = S (Ut, t) ⊂ M × R as in Corollary 4.4. Denote by F : V 7→ U the map that sends a point

gt −1 (p, t) ∈ V , with p ∈ Ut, to (h ) (p) and by Ft the restriction of F to (Ut, t). For any point p ∈ U the preimage F −1(p) is a parameterized curve (hgt(p), t). Let

∂ w = (˜v, ∂t ) be the vector field on V that consists of all the tangent vectors to all the parameterized curves F −1(p). Chapter 4. Differentiation rule 67

Assume that Ω is a k-form on V ⊂ M × R the restriction of which to horizontal

∗ sections Ut × {t} ⊂ V coincide with the form Ft ω and such that i ∂ Ω = 0. Informally, ∂t ∗ the latter means that Ω does not contain the differential dt. The form Ft ω considered as a form on V satisfy the above requirements and is, in fact, the only such a form Ω.

Let us apply Liouville-type formula to the vector field w on V , the form Ω, and to the

d R image D of a k-dimensional disc with D ⊂ U × {0}. We obtain: dt V (t) = LwΩ. The Dt shift diffeomorphism at time s of the vector field w maps the zero section U × {0} into

the time s section Us × {s} and is the inverse of the restriction mapping Fs : Us × {s} 7→

R R ∗ R U × {0}. After the change of coordinates we have V (t) = Ω = Ft ω = ω. Dt −1 D0 (Ft )∗D0 ∗ We conclude that 0 = LwΩ = L ∂ Ω + Lv˜Ω. Now, I claim thatv ˜ = (Ft g)v. Indeed, in the ∂t system of coordinates in which the vector field v is ∂ the parametrized curve (hgt, t) is ∂x1

given by the formula (x1 + tg, x2, . . . , xn, t). Thus at the point (P, t) ∈ M × R the vector v˜ is g(Q) ∂ with hgt(Q) = P . It confirms the formulav ˜ = (F ∗g)v. ∂x1 t

As the integral curves of the vector fieldv ˜ belong to the horizontal sections (Ut, t), one

can compute Lv˜Ω by restricting the form Ω to the horizontal sections (Ut, t). Substitution

∗ ∗ ofv ˜ = (Ft g)v into Lv˜Ω|(Ut,t) = Lv˜Ft ω|(Ut,t) and the use of the identity Lfvα = Lvfα, which is valid for any vector field v, any function f, and any differential form α of

∗ the highest degree on a manifold, yields to Lv˜Ω = LvFt (gω). Finally, note that for a

multivector ξ at a point p ∈ U the expression L ∂ Ω evaluated on the multivector ξ at ∂t d gt −1 the point (p, t) is dt [ω((h )∗ ξ). This completes the proof.

Remark 4.3. The analysis of the proof shows that it is sufficient to assume all the data

in Proposition 4.6 to be C1-smooth.

The following particular cases of Proposition 4.6 and Corollary 4.7 will be important

in the proof of the converse Abel’s theorem. Suppose that f = (f1, . . . , fn): M 7→ U is a diffeomorphism of a real manifold M into a domain U ⊂ Rn. Let g : M 7→ R be a function, ω be a differential form on M of the highest degree, and v be a constant vector Chapter 4. Differentiation rule 68

n n n field in R that corresponds to a vector v ∈ R . Consider the mapping fa : M 7→ R

given by the formula fa = f + agv, where a ∈ R. By Corollary 4.4, locally the mapping

−1 −1 ∗ fa is well-defined. Denote the form (fa ) ω by tracefa ω. Let [tracefa ω] and [tracefa gω] be the functions defined from the identities tracefa ω = [tracefa ω]Ω and tracefa gω =

n [tracefa gω]Ω, where Ω is the standard volume form on R . Corollary 4.4 together with the direct application of Proposition 4.6, and Corollary 4.7 brings us to the following two corollaries, respectively.

S Corollary 4.8. There is an arbitrarily small open connected neighborhood V = (Ua, a) of any prescribed point (p, 0) ∈ U × R with fa : U0 7→ Ua a diffeomorphism of connected open sets, such that the functions [tracefa ω] and [tracefa gω] are well-defined on V and satisfy the following partial differential equation:

∂ [trace ω] = −L [trace gω]. ∂a fa v fa

In particular, for any constant vector field v the derivative of [tracefa ω] with respect to

∂ a at a = 0 is well-defined on the whole domain U and ∂a [tracefa ω]|a=0 = −Lv[tracef gω].

Corollary 4.9. It is true that

∂s [trace ω]| = (−1)sLs[trace gsω], ∂as fa a=0 v f

where s ∈ N. Chapter 5

Main theorems: Converse of Abel’s

theorem - Polynomial case

Unless otherwise stated, the manifolds, mappings, and forms considered below are as-

sumed sufficiently smooth. Let M and N be real manifolds and f : M 7→ N is a finite

cover of degree d. Each point of P of N has a neighborhood U whose preimage contains

d connected components U1,...,Ud such that f : Uk 7→ U is a diffeomorphism. Denote

by gk the inverse of the mapping f|Uk . Let ω be a form on M. The trace of ω is the form d P ∗ traceπω = gkω. k=1 n+k n k Let γ1, . . . , γd be n-dimensional surfaces in R = R × R that are the graphs of

1 k n C vector functions gi : U 7→ R defined in a connected domain U of R . Let π be the

projection of Rn+k onto the first multiplier. The k-dimensional plane π−1(0) is called vertical.

Assume that, for any k-dimensional plane L that is sufficiently close to the vertical

n one, there is a connected domain UL in R that differs fairly little from U so that the

n parallel projection πL of γ1, . . . , γd onto R along L defines a d-sheeted covering of the −1 S preimage πL (UL) ⊆ γi onto UL. We first state the converse of Abel’s theorem for

hypersurfaces, t.e when k = 1. Denote the standard volume form on Rn+1 by Ω. Below

69 Chapter 5. Main theorems: Converse of Abel’s theorem - Polynomial case70

it is our convention that the polynomial identically equal to zero has the degree −1.

1 Theorem 5.1 (Converse of Abel’s theorem 1). Let ω1, . . . , ωd be C forms of the highest

degree on γ1, . . . , γd that vanish on nowhere dense subsets. Then for any line L that

is sufficiently close to the vertical one, the trace of ω1, . . . , ωd under the mapping πL is

a polynomial form PLdx1 ∧ ... ∧ dxn on UL and the degrees of the polynomials PL are bounded by a single constant N if and only if

1. In Rn+1, there is a (singular) n-dimensional algebraic variety Γ of degree d that

contains γ1, . . . , γd.

2. On Γ, which is defined by a polynomial P without multiple irreducible factors, there

QΩ is a form of the type dP where degQ ≤ N + d − 1, the restriction of which to each

surface γk coincide with ωk.

I recall that a k-form ω on a k-dimensional algebraic variety Γ ⊂ Cn is generalized holomorphic if ω is holomorphic on the non-singular part of Γ, and the trace of ω under

any proper finite polynomial mapping f :Γ 7→ Ck is a polynomial k-form on Ck (see page 60).

1 Theorem 5.2 (Converse of Abel’s theorem 2). Let ω1, . . . , ωd be C forms of the highest

degree on γ1, . . . , γd that vanish on nowhere dense subsets. Then for any k-dimensional

plane L that is sufficiently close to the vertical one, the trace of ω1, . . . , ωd under the

mapping πL is a polynomial form PLdx1 ∧... dxn on UL and the degrees of the polynomials

PL are bounded by a single constant if and only if

1. In Rn+k, there is a (singular) n-dimensional algebraic variety Γof degree d that

contains γ1, . . . , γd.

2. On the complexification of the variety Γ, there is a generalized holomorphic form

of the highest degree the restriction of which to each surface γk coincide with ωk. Chapter 5. Main theorems: Converse of Abel’s theorem - Polynomial case71

Remark 5.1. We say that the domain UL differs fairly little from U if there is a small

neighborhood Up of any prescribed point p ∈ U such that the domains Up ⊂ UL for the k-dimensional plane L that is sufficiently close to the vertical one.

Remark 5.2. Interestingly, as the analysis of our proof unravels, in the condition of

Abel’s theorem 5.1 it is sufficient to require that there is only one point p ∈ U such that

−1 ω1, . . . , ωd do not vanish at π (p).

Remark 5.3. In the condition of the converse of Abel’s theorem it is sufficient to require

1 that γ1, . . . , γd be the graphs of C vector functions and that the forms ω1, . . . , ωd be continuous.

The converse of Abel’s theorem has a complex-analytic version, stated below, in which

n+k n+k R is replaced by C and the surfaces γ1, . . . , γd and the forms ω1, . . . , ωd are analytic. This version extends Griffiths’ theorem, and its proof coincide nearly word for word with the proof of the real version.

n+k n k Let γ1, . . . , γd be n-dimensional surfaces in C = C × C that are the graphs of

k n holomorphic vector functions gi : U 7→ C defined in a connected domain U of C . Let

π be the projection of Cn+k onto the first multiplier. The k-dimensional plane π−1(0) is called vertical.

Assume that, for any k-dimensional plane L that is sufficiently close to the vertical

n one, there is a connected domain UL in C that differs fairly little from U so that the

n parallel projection πL of γ1, . . . , γd onto R along L defines a d-sheeted covering of the −1 S preimage πL (UL) ⊆ γi onto UL. We first state the converse of Abel’s theorem for hypersurfaces, t.e when k = 1. Denote the standard holomorphic volume form on Cn+1 by Ω.

Theorem 5.3. Let ω1, . . . , ωd be holomorphic forms of the highest degree on γ1, . . . , γd that do not vanish identically. Then for any line L that is sufficiently close to the vertical one, the trace of ω1, . . . , ωd under the mapping πL is a polynomial form PLdz1 ∧ ... ∧ dzn Chapter 5. Main theorems: Converse of Abel’s theorem - Polynomial case72

on UL and the degrees of the polynomials PL are bounded by a single constant N if and only if

1. In Cn+1, there is a (singular) n-dimensional algebraic variety Γ of degree d that

contains γ1, . . . , γd.

2. On Γ, which is defined by a polynomial P without multiple irreducible factors, there

QΩ is a form of the type dP where degQ ≤ N + d − 1, the restriction of which to each

surface γk coincide with ωk.

Theorem 5.4. Let ω1, . . . , ωd be holomorphic forms of the highest degree on γ1, . . . , γd that do not vanish identically. Then for any k-dimensional plane L that is sufficiently

close to the vertical one, the trace of ω1, . . . , ωd under the mapping πL is a polynomial

form PLdz1 ∧... dzn on UL and the degrees of the polynomials PL are bounded by a single constant if and only if

1. In Cn+k, there is a (singular) n-dimensional algebraic variety Γof degree d that

contains γ1, . . . , γd.

2. On Γ, there is a generalized holomorphic form of the highest degree the restriction

of which to each surface γk coincide with ωk.

Remark 5.4. In the view of Lemma 6.2 (see Proposition 2.20 for a more general statement) the degrees of the polynomials PL are automatically bounded by a single constant. Chapter 6

Proofs of Main theorems

Throughout this chapter we use the notations from the chapter 5.

6.1 Sufficiency: Theorem 5.1 and Theorem 5.2

For the proof of sufficiency in Theorem 5.1 we will need the following lemma.

Let P ∈ R[x][y] be a monic polynomial in a variable y whose coefficients are poly- nomials in variables x = (x1, . . . , xn). Then for an arbitrary polynomial Q ∈ R[x][y] the remainder R ∈ R[x][y] of the division Q by P is well-defined. The degree of R as a polynomial in variable y is strictly less the degree of Q.

Lemma 6.1. Assume that the total degree of each summand in the expression P =

d d−1 y + b1y + ... + bd is at most d. Then the total degree of the remainder R is at most as the total degree of the polynomial Q.

Proof of Lemma 6.1. It is a straightforward computation to show that the statement of the lemma remains valid at each step of the division algorithm.

Proof of Theorem 5.1. Sufficiency. Let Γ ⊂ Rn × R be an algebraic hypersurface of QΩ degree d defined by a polynomial P without multiple irreducible factors and ω = dP a residue form on Γ with degQ ≤ N + d − 1. Assume that πL is a parallel projection

73 Chapter 6. Proofs of Main theorems 74

n T −1 of Γ onto R along a line L that defines a d-sheeted covering of Γ π (UL) over an

n open subset UL ⊂ R . We will prove that traceπL ω = PLdx1 ∧ ... ∧ dxn, where PL is a polynomial of degree at most N.

n Let (x1, . . . , xn, y) = (x, y) be coordinates in R × R in which L is the vertical line

d d−1 x = 0. In these coordinates ,up to multiplication by a constant, P = y +b1y +...+bd and the degree of each summand is at most d. Now, up to multiplication by a constant,   Qdx1∧...∧dxn Q Q ω = 0 and traceπL ω = 0 (p1) + ... + 0 (pd) dx1 ∧ ... ∧ dxn, where p1, . . . , pd Py Py Py −1 are the preimages πL (p) of a point p ∈ UL. Q Q Q By the residue theorem, the sum 0 (p1) + ... + 0 (pd) is equal to Res∞ dy with the Py Py P P Q the function Q restricted to the vertical line x = x(p). To compute Res∞ P |x=x(p)dy, we

write Q as a polynomial in y with coefficients in R[x1, . . . , xn], compute the remainder

d−1 R of the division of Q by P and take the coefficient R0 before y . If N 6= −1, then by

Lemma 6.1, the degree of the polynomial R0 ∈ R[x1, . . . , xn] is at most N; if N = −1,

d−1 then degQ ≤ d − 2 so the reminder R0 = Q and, again, the coefficient R0 before y is equal to zero.

The preimage of a point under the projection of Rn × Rk onto the second multiplier is an

n-dimensional plane in Rn ×Rk. Such a plane we call horizontal. The proof of sufficiency in the theorem 5.2 is the straightforward corollary from the following

Lemma 6.2. Let f : U 7→ R be a real analytic function defined on a connected domain

U in Rn × Rk. Suppose that that for any n-dimensional horizontal plane l the restriction of f|l T U is a polynomial. Then the degrees of all the polynomials are uniformly bounded from above.

n Proof of Lemma 6.2. Let y = (y1, . . . , yn) be coordinates in R . It is sufficient to show ∂αf that in a neighborhood of a point in U all the derivatives ∂yα vanish starting with some order N. To simplify our notations we will provide the proof for n = 1. Chapter 6. Proofs of Main theorems 75

Choose a closed disk V ⊂ Rk and a closed interval ∆ ⊂ R so that ∆ × V ⊂ U. In

2 ∆×V the function f admits the following representation f = c0 +c1y +c2y +..., where y is a coordinate on the line R and ck are analytic functions on V . Let Sm ⊂ V be the

set of points p so that cl(P ) = 0 for all l ≥ m. The sets Sm are closed and their union is

V . Since V is a Baire space there is an integer N and a point p ∈ V so that p ∈ SN is an interior point. In particular, in a sufficiently small neighborhood of p all cl, starting

∂αf with l ≥ N, vanish. We conclude that cl, which is equal to ∂yα , is identically zero on the domain U if l ≥ N.

Proof of Theorem 5.2. Necessity. We start with the graphs γ1, . . . , γd that belong to an

n+k n k algebraic variety Γ ⊂ R = R × R of degree d and forms ων that are the restriction of a generalized holomorphic form ω on Γ to γν.

k ∗ Let l1, . . . , lk be a fixed basis in (R ) . A parallel projection along a k-dimensional plane L is of the form π + l1v1 + ... + lkvk, where v1, . . . , vk are sufficiently small vectors

n n k nk in R uniquely determined by L. Let v = (v1, . . . , vk) ∈ (R ) = R . We denote the domain UL by Uv and the projection along a k-dimensional plane L by πLv .

n nk The collection of points (Uv, v) ⊂ R × R is a connected subset, which we denote by V . Any point of V correspond to a k-dimensional plane which transversely meets the surfaces γ1, . . . , γd. It follows that there is a connected open set U containing V with the property that each point of V corresponds to a k-dimensional plane meeting the surfaces γ1, . . . , γd transversely. Let ρ0 be the function on U defined from the identity

n traceπLv ω1 + ... + traceπLv ωd = ρ0Ω0, where Ω0 is the standard volume form on R . By the assumption, for any fixed v ∈ U ⊂ Rnk the function ρ is a polynomial. The application of Lemma 6.2 to the domain U ⊂ Rn × Rnk completes the proof.

6.2 Necessity: Theorem 5.1

We start with the proposition that is a “toy-model” of our main theorems and demon- Chapter 6. Proofs of Main theorems 76

strates the idea behind the proof of algebraizabilty-type results. Let γ1, . . . , γd be n-

dimensional surfaces in Rn+1 = Rn × R that are the graphs of (arbitrary, even not

necessarily continuous) real-valued functions defined on a connected domain U of Rn.

Suppose that π, the projection of Rn+1 onto the first multiplier, induces a d-sheeted S covering π : γk 7→ U. Let a coordinate y be fixed on the line R. Denote by yj the

restriction of y to γj.

Proposition 6.3. Let ρ1, . . . , ρd be arbitrary functions on γ1, . . . , γd that vanish nowhere.

k Assume that, for any non-negative k that is at most 2d − 1, the function sk = ρ1y1 +

k ... + ρdyd is a polynomial on U. Then:

1. There is a polynomial F in Rn+1 whose zero locus in U × R as well as in U × C is

the union of all the surfaces γν. The degree of F in the variable y is d.

2. There is a rational function P/Q in Rn+1 with Q vanishing at no point of the

surfaces γν such that P/Q restricted to each γν coincide with ρν.

Proof of Proposition 6.3. The proof immediately follows from Theorem 2.8 with M = S γk, N = U, g = y, and the function ρ equal to ρk when restricted to γk. Indeed, the

S d union γk ⊂ U × R is defined by the equation Sd(s)y + ... + S0(s) = 0 with Sk ∈

Z[x1, . . . , x2d] universal, in the sense of Theorem 2.8, polynomials and s = (s0, . . . s2d−1);

1 2d−1 the function ρ = (T2d−1(s)y + ...T0(s)) with Tk ∈ Z[x1, . . . , x2d] universal, in T2d(s) the sense of Theorem 2.8, polynomials. Finally, note that for a fixed point p ∈ U the

d polynomial Sd(s(p))y + ... + S0(s(p)) = 0 of one variable has only real roots.

Remark 6.1. As follows from the proof of Theorem 2.8, the polynomial F in Proposi- Chapter 6. Proofs of Main theorems 77

tion 6.3(1) has a particularly beautiful form:   s s . . . s  0 1 d     s s . . . s   1 2 d+1     . . .. .  det  . . . .      sd−1 sd . . . s2d−1     1 y . . . yd

Proof of Theorem 5.1. Necessity. Suppose that the trace of ω1, . . . , ωd under the mapping

πL is a polynomial form PLdx1 ∧ ... ∧ dxn on UL and the degrees of the polynomials PL are bounded by a single constant N.

In Step 1, to demonstrate our methods, we show the existence of the algebraic variety of degree d when γ1, . . . , γd and ω1, . . . , ωd are sufficiently smooth, and the forms vanish nowhere on the graphs. In Step 2, assuming the existence of the algebraic variety of

QΩ degree d, we show the existence of the residue form dP , with degQ ≤ N + d − 1, the

restriction of which to each graph γν coincide with ων. In Step 3, we show the existence of the algebraic variety of degree d when the graphs and the forms are C1 smooth, and

the forms vanish nowhere on the graphs. Finally, in Step 4, we extend the arguments of

Step 3 to allow the forms ων to vanish on nowhere dense subsets of γν.

Step 1. Let a coordinate y be fixed on the line R, v be an arbitrary vector in Rn, and

n n+1 n Ω0 = dx1 ∧ ... ∧ dxn be the standard volume form in R . Recall that π : R 7→ R

n+1 n is the projection along a vertical line. Consider the family of mappings πa : R 7→ R

defined by the formula πa = π + ayv, where a ∈ R. For sufficiently small a, the mapping

n πa is the projection along a straight line `a close to a vertical one. For each vector v ∈ R ,

applying Corollary 4.9, we compute the derivative of traceπa ω with respect to a at a = 0.

This computation shows that the trace of yω1, . . . , yωd under the projection π = π0 is

polynomial P1dx1 ∧ ... ∧ dxn with degP1 ≤ N + 1. Similarly, computing the higher

derivatives of traceπa ω with respect to a at a = 0, we find that, for any positive integer

k k, the form traceπa y ω is polynomial Pkdx1 ∧ ... ∧ dxn with degPk ≤ N + k. Chapter 6. Proofs of Main theorems 78

Denote by ρi the functions on U defined from the identities traceπωi = ρiΩ0 and by yi

the restriction of the coordinate function y to γi. It follows that for every positive k the

k−1 k−1 n function sk = ρ1y1 + ... + ρdyd is a polynomial of degree at most N + k − 1 on R .

By Proposition 6.3, there is a polynomial in Rn+1 whose zero locus in U × R as well as in U × C is exactly γ1, . . . , γd. In particular, the graphs and the forms are real-analytic. Now we show that there is an algebraic hypersurface of degree precisely d that contains the graphs. Indeed, let Γ be smallest of algebraic varieties that contains γ1, . . . , γd. By applying the above argument for the projection along a straight line ` that is sufficiently

n+1 close to the vertical one, we find a polynomial in R whose zero locus Γ1 has the following two properties: 1) the intersection of Γ1 with the complexification of each line

−1 from the family of parallel lines {π` (p)|p ∈ U`} consists of exactly d points. 2) Γ ⊂ Γ1. It follows that Γ is of degree d.

d−1 Step 2. Consider the polynomial Q = ad−1y + ... + a0 whose coefficients are

polynomials in Rn and are related to the coefficients of the defining Γ polynomial P =

d d−1 y + b1y + ... + bd by the following formula

      ad−1 s1 0 0 ... 0 1             ad−2 s2 s1 0 ... 0   b1    =     .  .   . . . . .   .   .   ......   .              a0 sd sd−1 s2 . . . s1 bd−1

QΩ Theorem 2.13 show that the restriction of the form dP to the graph γk coincide with

ωk. Since bi is a polynomial of degree at most i (1 ≤ i ≤ d − 1) and sj is a polynomial of degree at most N + j − 1 (1 ≤ j ≤ 2d − 1), the polynomial Q has the degree at most

N + d − 1.

Step 3. It is sufficient to prove Theorem 5.1 in a sufficiently small neighborhood U0 of a prescribed point in U. By Proposition 4.5 (or by Proposition 6.7), locally, it is valid

to apply Proposition 6.5 below. Proposition 6.5 with g = y and the forms ω1, . . . , ωd Chapter 6. Proofs of Main theorems 79

gives that the trace of gω1, . . . , gωd under the mapping πL is a polynomial form on U0 if the line L is sufficiently close to the vertical. Proposition 6.5 with g = y and the forms

2 2 gω1, . . . , gωd gives that the trace of g ω1, . . . , g ωd under the mapping πL is a polynomial

form on U0 if the line L is sufficiently close to the vertical. Continuing in the same way and substituting a = 0, we again obtain that, for every non-negative k, the sum

k k n sk|U0 = (ρ1y1 + ... + ρdyd )|U0 is a polynomial on R . From here we follow the arguments of Step 1 and Step 2.

Step 4. The arguments of the previous step show that the functions sk are locally

polynomials on U. Since U is connected, we conclude that each sk coincide with a single polynomial on U.

Now choose a neighborhood U0 ⊂ U of a point in U so that the forms ωk do not

d vanish on γk|U0 . Let P = ∆0y + ... + ∆d be the polynomial from Remark 6.1, where the coefficients ∆k ∈ R[x1, . . . , xn]. By Step 1, the coefficients of P/∆0 are polynomials

n+1 on U0, and P/∆0 is a polynomial of degree d in the ambient space R . According to

Proposition 2.13, in U ×R the zero locus of P/∆0 outside the (an algebraic) hypersurface

n+1 Σ = {∆0 = 0} ⊂ R is γ1|U\Σ, . . . , γd|U\Σ. By continuity, the polynomial P/∆0 vanishes on all the surfaces γν.

Proposition 6.5 below shows that, under an extra assumption, forms that satisfy the

polynomial trace condition in the necessity of Theorem 5.2 constitute a module over the

ring of polynomials with real coefficients. Proposition 6.7 explains that locally the extra

assumption in Proposition 6.5 is satisfied.

Suppose that γ is an n-dimensional surface in Rn+k = Rn × Rk that is the graph of

1 k n a C vector function g : U 7→ R defined in a connected domain U of R . Let l1, . . . , lk

k ∗ n k be a fixed basis in (R ) . A parallel projection πL of the surface γ in R × R along a k-dimensional plane L is of the form π + l1v1 + ... + lkvk, where v = (v1, . . . , vk) is a k-tuple of sufficiently small vectors in Rn uniquely determined by L. Denote the parallel Chapter 6. Proofs of Main theorems 80

n k n n k projection πL by πLv , and consider the mapping γ × (R ) 7→ R × (R ) that sends a

point (p, v) to (πLv (p), v). The next proposition is the direct corollary from the implicit function theorem.

˜ S ˜ Proposition 6.4. There is an arbitrarily small open connected neighborhood U = (Uv, v) of any prescribed point (p, 0) ∈ U × ( n)k so that π : γ| 7→ U˜ is a diffeomorphism, R Lv U˜0 v where γ| is the restriction of the surface γ to the subdomain U ⊂ U. U˜0 0

Now suppose that ω1, . . . , ωd are n-forms on γ1, . . . , γd that satisfy the polynomial trace condition with the constant N in the necessity of Theorem 5.2. Denote the domain

n n k UL by Uv. The collection of points (Uv, v) ⊂ R × (R ) over all k-tuples of sufficiently small vectors (v1, . . . , vk) is a connected subset, which we denote by V . We do not assume anything on the zero locus of the forms, however, suppose that S V = (Uv, v) is a connected domain. Under these assumptions the following proposition is valid.

Proposition 6.5. Let g be a polynomial with real coefficients in Rn+k = Rn × Rk. Then for any k-dimensional plane L that is sufficiently close to the vertical one, the trace of gω1, . . . , gωd under the mapping πL is a polynomial form PLdx1 ∧ ... dxn on UL and the degrees of the polynomials PL are bounded above by N +degg. Here the surfaces γ1, . . . , γd

1 and the forms ω1, . . . , ωd are assumed to be C smooth.

Proof of Proposition 6.5. It is sufficient to consider the case when g is a linear polynomial.

n n+k n Take any vector v in R , and consider the family of mappings πa : R 7→ R defined by the formula πa = πL + agv, where a ∈ R. For sufficiently small a, the mapping πa is, up to multiplication by a linear function in a, a projection along a k-dimensional plane close to L.

For each vector v ∈ Rn, applying the corollary 4.8, we compute the derivative of traceπa ω = traceπa ω1 + ... + traceπa ωd with respect to a at a = 0. Locally, the

P α function [traceπa ω] = cαx , where α = (α1, . . . , αn) is a multi-index, cα : I 7→ Chapter 6. Proofs of Main theorems 81

R are C1 smooth functions of a parameter a defined on interval I of the real line,

α α1 αn and x = x1 ··· xn . We denote traceπa gω1 + ... + traceπa gωd by traceπa gω. Since

∂ −Lv[traceπ0 gω] = ∂a [traceπa ω]|a=0 for every constant vector field v, we conclude that the function [traceπa gω]|a=0 = [traceπL gω], which is defined on UL, is a polynomial of degree at most N + 1.

Remark 6.2. Above we used the following self-evident fact

Let f be a polynomial in n variables with coefficients in the ring of all complex-valued

functions on a connected domain V ⊂ Rm. Assume that for a non-negative integer k the

function f : V × U 7→ C, where U is a domain in Rn, is a Ck (resp. analytic) function of the first argument. Then all the coefficients of f are Ck (resp. analytic) functions on V .

6.3 Necessity: Theorem 5.2

To prove the existence of a generalized holomorphic form as claimed in the sufficiency of

the theorem 5.2 we will need the following lemma, which we prove in Chapter 10.

Lemma 6.6. Let Γ be an n-dimensional complex algebraic variety in an affine space and

ω an n-form on Γ holomorphic outside of a hypersurface that contains the singular locus

of Γ. Then the following three conditions on the form ω are equivalent:

1. There is a proper finite polynomial mapping f :Γ 7→ Cn so that for any polynomial

g with complex coefficients the trace of gω under the mapping f is Pgdy1 ∧ ... ∧ dyn

n with Pg ∈ C[y1, . . . , yn] a polynomial in a fixed affine coordinates y1, . . . , yn in C .

2. The trace of ω under any proper finite polynomial mapping F :Γ 7→ Cn is of the

form PF dy1 ∧...∧dyn with PF ∈ C[y1, . . . , yn] in a fixed affine coordinates y1, . . . , yn

in Cn. Chapter 6. Proofs of Main theorems 82

Proof of Theorem 5.2. Sufficiency. In Step 1 we show the existence of an algebraic variety

of degree d that contains γ = (γ1, . . . , γd). We show that by, locally, projecting the surfaces into the affine space of lower dimension and then applying Theorem 5. In Step

2 we show the existence of a generalized holomorphic form ω which restriction to each

γk is ωk.

n Step 1. We use Proposition 2.16 to locally project the surfaces πj : γk 7→ R × R one- to-one on their images πj(γν), where j = 1, . . . , k. For a fixed projection πj, consider the

−1 ∗ n n forms (πj ) ωk on the graphs πj(γk) in R × R. If ` is a line in R × R that is sufficiently

−1 n k close to the vertical, then its preimage πj (`) ⊂ R × R is a k-dimensional plane L

−1 ∗ sufficiently close to the vertical. Since, for each surface γk, the trace traceπ` (πj ) ωk = traceπL ωk, Theorem 5 asserts that the surfaces πj(γ1), . . . , πj(γd) belong to algebraic hypersurface Γπj of degree d. Considering all the k projections π1, . . . πk, it follows that locally the surfaces belong to an n-dimensional algebraic variety, and (also locally) there is a rational n-form on the variety which restriction to each γν is ων. In particular, all the surfaces γ1, . . . , γd are the graphs of (real) analytic vector-functions.

Apply the same arguments to the projections πL along k-dimensional planes that

are close to the the vertical one. Consider all the hypersurfaces Γπj as hypersurfaces in

Rn+k. Their intersection Γ is an algebraic variety in Rn+k that, due to analyticity of the surfaces, contains γ1, . . . , γd. Without loss of generality, we may assume that Γ is an n-dimensional algebraic variety. I claim that the degree of Γ is d. Indeed, for a generic, in the sense of Zariski topology, projection πj the image of Γ is a hypersurface of degree d. Thus Γ is an algebraic variety of degree d.

Note that, by Proposition 6.3, there is a rational n-form ω on Γ which restriction to each γν is ων. We will employ this in Step 2.

Step 2. Let Γ ⊂ Rn+k = R × Rk be an algebraic variety of degree d that contains

γ1, . . . , γd and ω be a rational form of the highest degree which restriction to each γν is ων.

Let Γc ⊂ Cn+k = Cn × Ck be the complexification of Γ, and ωc be the complexification Chapter 6. Proofs of Main theorems 83

of ω. The complexification πc of the projection π, by Noether normalization lemma,

induces a proper finite polynomial mapping of Γc onto Cn. The result of Proposition 6.7 ensures that, locally, we can apply Proposition 6.5. Thus

for any polynomial g ∈ R[x1, . . . , xn+k] with real coefficients the trace of gω under the

projection π :Γ 7→ Cn is a polynomial form on U. It follows that, for any polynomial

c c n g ∈ C[x1, . . . , xn+k], the trace of gω under the projection π :Γ 7→ C is a polynomial

form on Cn. The application of Lemma 6.6 completes the proof.

We finish this section with the statement of the complex-analytic analogues of Proposi-

tions 6.7 and 6.5. We will employ them in the appendix and also in the characterization

of generalized holomorphic forms on complete intersetions.

Suppose that γ is an n-dimensional surface in Cn+k = Cn × Ck that is the graph of

an analytic vector function g : U 7→ Ck defined in a connected domain U of Cn. Let

k ∗ n k l1, . . . , lk be a fixed basis in (C ) . A parallel projection πL of the surface γ in C × C along a k-dimensional plane L is of the form π + l1v1 + ... + lkvk, where v = (v1, . . . , vk) is a k-tuple of sufficiently small vectors in Cn uniquely determined by L. Denote the

n k n n k parallel projection πL by πLv , and consider the mapping γ × (C ) 7→ C × (C ) that sends a point (p, v) to (πLv (p), v). The next proposition is the direct corollary from the implicit function theorem.

˜ S ˜ Proposition 6.7. There is an arbitrarily small open connected neighborhood U = (Uv, v) of any prescribed point (p, 0) ∈ U × ( n)k so that π : γ| 7→ U˜ is a biholomorphic C Lv U˜0 v isomorphism, where γ| is the restriction of the surface γ to the subdomain U ⊂ U. U˜0 0

Now suppose that ω1, . . . , ωd are n-forms on γ1, . . . , γd that satisfy the polynomial trace condition with the constant N in the necessity of Theorem 5.4. Denote the domain

n n k UL by Uv. The collection of points (Uv, v) ⊂ C × (C ) over all k-tuples of sufficiently small vectors (v1, . . . , vk) is a connected subset, which we denote by V . Chapter 6. Proofs of Main theorems 84

We do not assume anything on the zero locus of the forms, however, suppose that S V = (Uv, v) is a connected domain. Under these assumptions the following proposition is valid.

Proposition 6.8. Let g be a polynomial with complex coefficients in Cn+k = Cn × Ck. Then for any k-dimensional plane L that is sufficiently close to the vertical one, the trace of gω1, . . . , gωd under the mapping πL is a polynomial form PLdx1 ∧ ... dxn on UL and the degrees of the polynomials PL are bounded above by N + degg. Chapter 7

Applications of Main theorems and our methods

7.1 Preparatory Theorems for Chapter 9

Our first application is Theorem 7.1 which we need for classification of hypersufaces of double translation in chapter 9.

n 1 n Let γk : ∆k 7→ R (k = 1, . . . , d) be C -smooth parameterized curves in R that are defined on intervals ∆k ⊂ R with the mappings γk being injections and dγk non- zero on ∆k. Below we will abuse the notations by referring to the image of ∆k as to the curve γk. Suppose that for each point on a curve γk there is a hyperplane L0 such that any hyperplane L that is sufficiently close to L0, in the sense of topology on the space of hyperplanes (RP n)∗, transversely meets the curves at precisely d points T T γ1 L, . . . , γd L.

2 Theorem 7.1. Let f1, . . . , fd be C functions on γ1, . . . , γd such that their differentials dfk vanish on nowhere dense subsets. Assume that for each hyperplane L meeting the T curves γ1, . . . , γd transversely at d points Pν = γν L the following identity is valid:

f1(P1) + ... + fd(Pd) = 0.

85 Chapter 7. Applications of Main theorems and our methods 86

Then:

n+1 1. In R , there is a (singular) algebraic curve Γ of degree d that contains γ1, . . . , γd.

2. On the projective closure of the complexification of Γ there is a generalized holo-

morphic form the restriction of which to each curve γk coincide with dfk.

We also state the complex-analytic analogue if Theorem 7.1 above. We will employ it

when we discuss Torelli theorem.

n n Let γk : ∆k 7→ C (k = 1, . . . , d) be analytic parameterized curves in R that are

defined on connected and simply-connected domains ∆k ⊂ C with the mappings γk being injections and dγk non-zero on ∆k. Below we will abuse the notations by referring to the image of ∆k as to the curve γk. Suppose that for each point of a curve γν there is a

hyperplane L0 such that any hyperplane L that is sufficiently close to L0, in the sense of topology on the space of hyperplanes (CP n)∗, transversely meets the curves at precisely T T d points γ1 L, . . . , γd L.

Theorem 7.2. Let f1, . . . , fd be holomorphic functions on γ1, . . . , γd such that their dif- ferentials dfk do not vanish identically. Assume that for each hyperplane L meeting the T curves γ1, . . . , γd transversely at d points Pν = γν L

f1(P1) + ... + fd(Pd) = 0.

Then:

n+1 1. In C , there is a (singular) algebraic curve Γ of degree d that contains γ1, . . . , γd.

2. On the projective closure of Γ there is a generalized holomorphic form the restriction

of which to each curve γk coincide with dfk.

Proof of Theorem 7.1. We first show the existence of the algebraic curve Γ. It is sufficient to show that locally. Fix a hyperplane L0 that meets γ1, . . . , γd transversely at d points T Pν = γν L0, and a line ` transversal to L0. Then near the intersection points P1,...,Pd Chapter 7. Applications of Main theorems and our methods 87

the curves γ1, . . . , γd are the graphs of vector-functions defined on a sufficiently small T interval ∆ ⊂ l containing ` L0. Applying our second main theorem to the forms

c n df1,..., dfd, we complete the proof. Also note that on the complexification Γ ⊂ C of Γ there is a generalized holomorphic for ω the restriction of which to each curve γk coincide

with dfk.

c n ∗ Let L0 be the complexification of L. Then any hyperplane L ⊂ (CP ) that is

c c sufficiently close to L0 meets Γ transversely in d points Pν at which

c c f1 (P1) + ... + fd (Pd) = 0,

c where fν is the complexification of fν. Since this condition is invariant under projective

n n c T n transformations, we conclude that, in any affine chart C ⊂ CP , the form ω|Γ C is generalized holomorphic.

7.2 Generalized holomorphic forms on complete in-

tersections in Cn and CP n

The next two theorems, essentially, summarize the previous results. Necessity and suf-

ficiency in Theorem 7.4 corresponds to Lemma 10.2 and Corollary 3.31, respectively.

Sufficiency in Theorem 7.3 corresponds to Corollary 3.32, while necessity follows from the formula in Remark 10.1.

n Theorem 7.3. Suppose that D1,...,Dk are algebraic hypersurfaces in CP that meet each other transversely at a generic point, in Zariski sense, of the (n − k)-dimensional T T projective subvariety Γ = D1 ... Dk. Let ω be an (n − k)-form holomorphic on the

non-singular locus of Γ. Assume that in the affine chart Cn ⊂ CP n with coordinates x1, . . . , xn the hypersurfaces D1,...,Dk are defined by polynomials P1,...,Pk. Then ω is Chapter 7. Applications of Main theorems and our methods 88

a generalized holomorphic form if and only if there is a polynomial Q ∈ C[x1, . . . , xn] of Qdx1 ∧ ... ∧ dxn the degree at most degP1 + ... + degPk − n − 1 so that ω = . dP1 ∧ ... ∧ dPk

Proof of Theorem 7.3. Necessity. We choose a k-dimensional plane L meeting Γ trans-

versely in d points P1,...,Pd, where d is the degree of the variety, and an (n − k)-

dimensional plane L0 that is transversal to L. By the implicit function theorem, near

the points Pν the variety Γ is the graph of an analytic vector functions defined on a

connected domain U ⊂ L0. We denote the parallel projection along L onto L0, as usual,

by πL. There is the natural affine structure in L0 induced by the affine structure in the

n n chart C ⊂ CP . Denote affine coordinates in L0 by y1, . . . , yn−k. By Proposition 6.7, for sufficiently small U, it is valid to apply Proposition 6.5, and

that implies that the trace of gω under the projection π is [traceπL gω]dy1∧...∧dyn−k with

[traceπL gω] ∈ C[y1, . . . , yn−k] a polynomial of degree at most deg g − 1 (I remind that, according to our convention, the identically zero polynomial has degree −1). It follows

Qdx1∧...∧dxn from Lemma 10.3 that there is a polynomial Q ∈ [x1, . . . , xn] so that ω = . C dP1∧...∧dPn−k Now we remind the formula for the polynomial Q established in Remark 10.1

X ∗ α Q = πL[traceπL cαω]x , |α|≤degH

where |α| = α1 + ... + αn and degcα ≤ degP1 + ... + degPn−k − n − |α|.

α Since the coefficient of Q before any of x with |α| = degP1 + ... + degPn−k − n is

zero and deg [traceπL gω] ≤ deg g − 1, we conclude that the degree of the polynomial Q is at most degP1 + ... + degPn−k − n − 1.

n Theorem 7.4. Suppose that D1,...,Dk are algebraic hypersurfaces in C that meet each other transversely at a generic point, in Zariski sense, of the (n − k)-dimensional affine T T subvariety Γ = D1 ... Dk. Let ω be an (n−k)-form holomorphic on the non-singular locus of Γ. Assume that in the affine coordinates x1, . . . , xn the hypersurfaces D1,...,Dk Chapter 7. Applications of Main theorems and our methods 89

are defined by polynomials P1,...,Pk. Then ω is a generalized holomorphic form if and Qdx1 ∧ ... ∧ dxn only if there is a polynomial Q ∈ C[x1, . . . , xn] so that ω = . dP1 ∧ ... ∧ dPk

7.3 Converse of Abel’s theorem - Rational case

The aim of this section is to prove Theorem 7.9.

We begin with a complex version of the “toy-model” in Proposition 6.3. Let γ1, . . . , γd be n-dimensional surfaces in Cn+1 = Cn ×C that are the graphs of holomorphic functions

n defined on a connected domain U of C . If U0 ⊂ U is a subdomain, we denote the

n+1 restriction of the graphs to U0 by γ1|U0 , . . . , γd|U0 . Suppose that π, the projection of C S onto the first multiplier, induces a d-sheeted covering π : γk 7→ U. Let a coordinate w be fixed on the line C. Denote by wj the restriction of w to γj. Below by a holomorphic function on a domain in Cn we mean an univalued function.

Proposition 7.5. Let ρ1, . . . , ρd be holomorphic functions on γ1, . . . , γd that do no vanish. Assume that there is a non-negative integer N so that the Nth order derivatives of the

k k functions sk = ρ1w1 + ... + ρdwd with 0 ≤ k ≤ 4d − 1 along any constant vector field in

Cn are rational on U. Then:

1. There is a polynomial P in Cn+1 whose zero locus in U × C is the union of all the

surfaces γν.

2. For a sufficiently small open disk U0 ⊂ U centered at a prescribed point p ∈ U,

there is a function R = (S0 + S1 ln P1 + ...Sm ln Pm)/Q with Pk,Sk,Q polynomials

n+1 on C and Q vanishing at no point of the surfaces γk|U0 so that each ln Pk is

holomorphic on γ1|U0 , . . . , γd|U0 and the restriction of R to each γν|U0 coincide with

ρν. The polynomials P1,...,Pm are polynomials in n complex variables defining the irreducible components of the polar locus of all the Nth order partial derivatives of

the functions s0, . . . , s4d−1. Chapter 7. Applications of Main theorems and our methods 90

Remark 7.1. Note that, given a holomorphic function f : U 7→ C defined on a connected

n domain U in C with a fixed affine coordinates z1, . . . , zn, the Nth order derivatives of f along any constant vector field in Cn are rational on U if and only if all the Nth order partial derivatives of f with respect to z1, . . . , zn are rational on U.

For the proof of Proposition 7.5(1) we need Lemma 7.6, which is well-known for n = 1.

We follow the notations of Proposition 7.5. Redefine U to be a small open disk centered at a point p of the domain U. Let K0 = C(z1, . . . , zn) be the field of rational

n functions on C , and K = K0(s0, . . . , s4d−1) be the field of meromorphic functions on

U obtained by the extension of K0 with all the branches of the multivalued functions s0, . . . , s4d−1. Let G be the monodromy group that acts on K by the analytic continuation n S S along any closed path α ⊂ C \(D1 ... Dm) that starts and ends at a point p ∈ U. Then

Lemma 7.6. The field K0 ⊂ K is the field of invariants under the G-action.

Proof of Lemma 7.6. We start with the case n = 1. Each sk satisfies a Fuchsian differ- ential equation. Thus the statement is a direct corollary from the Frobenius theorem

(see [10] or [17]). Now, by restricting to lines, the case n > 1 is a direct corollary from the case above, Proposition 2.21.

For the proof of Proposition 7.5(2) we need the following lemma, which is a simple

exercise for n = 1.

n Lemma 7.7. Suppose that D ⊂ C is an algebraic hypersurface with D1,...,Dm its

irreducible components, and f : U 7→ C is holomorphic on an open disc U ⊂ Cn\D. Assume that, for some non-negative integer N, all the Nth order partial derivatives of

f are rational on Cn. Then, on a maybe smaller open disc centered at the same point,

n f = R0 + S1 ln P1 + ...Sm ln Pm, where R0 is rational on C , Sk are polynomials on

n C , and P1,...Pm are irreducible polynomials defining the hypersurfaces D1,...,Dm, respectively. Chapter 7. Applications of Main theorems and our methods 91

Proof of Proposition 7.5. We prove the proposition for n = 1, and for an arbitrary n modulo Lemma 7.7 and Lemma 7.6. Redefine U to be a small open disk centered at a point p of the domain U. Let K0 = C(z1, . . . , zn) be the field of rational functions on

n C , and K = K0(s0, . . . , s4d−1) be the field of meromorphic functions on U obtained by the extension of K0 with all branches of the multivalued functions s0, . . . , s4d−1. Let G be the monodromy group that acts on K by the analytic continuation along any closed n S S path α ⊂ C \(D1 ... Dm) that starts and ends at a point p ∈ U. By Lemma 7.6, the field K0 ⊂ K is the field of invariants under the G-action. Let L ⊃ K be the field of meromorphic functions that is obtained from K by the extension with the elements

ρ1, . . . , ρd and w1, . . . , wd.

Now, since the action of the monodromy group is commutative with partial differen- tiation, if g ∈ G, then all the Nth order partial derivatives of the functions gsk coincide with the corresponding Nth order partial derivatives of the functions sk. Thus gsk − sk is a polynomial. The application of Proposition 2.18 completes the proof of Proposi- tion 7.5(1).

We proceed with Proposition 7.5(2). We work over the same open disk U centered at the point p. By Lemma 7.7 and Remark 7.1, each sk on a, maybe smaller, open disk U0 centered at p ∈ U has the form R0 +S1 ln P1 +...Sm ln Pm with S1,...Sm ∈ C[z1, . . . , zn].

Let P be the polynomial on Cn+1 as in Proposition 7.5. We write the polynomial P as

d d−1 y + b1y + ... + bd, where the coefficients are rational functions in z1, . . . , zn. Now

d−1 consider the polynomial Q = ad−1y + ... + a0 whose coefficients are holomorphic

d d−1 functions on U0 ⊂ C and are related to the coefficients of P = y + b1y + ... + bd by the following formula Chapter 7. Applications of Main theorems and our methods 92

      ad−1 s1 0 0 ... 0 1             ad−2 s2 s1 0 ... 0   b1    =     .  .   . . . . .   .   .   ......   .              a0 sd sd−1 s2 . . . s1 bd−1

QΩ Formulas from the theorem 2.13 show that the restriction of the form dP to the graph

γk coincide with ωk. This completes the proof of Proposition 7.5(2).

Let M be a complex manifold. We denote a pair of a purely dimensional analytic set

Γ ⊂ M together with a top degree form ω meromorphic on Γ by (Γ, ω). If (Γ1, ω1) and

(Γ2, ω2) are two such pairs with Γ1 and Γ2 of the same dimension, then (Γ1, ω1) +(Γ2, ω2) is, by definition, the set-theoretical union of Γ1 and Γ2 together with the form ω12 =

indΓ1 ω1 + indΓ2 ω2, where indΓk (k = 1, 2) is the indicator function of the analytic set Γk.

n+1 n We need the following construction. Let Γexc ⊂ C = C × C be the union of a

finite number of hyperplanes passing through an (n − 1)-dimensional plane A ⊂ Cn+1, and ` be a linear polynomial that defines yet another hyperplane passing through the

same plane A. Suppose that the parallel projection π of Cn+1 = Cn × C onto the first −1 T d n multiplier induces a finite covering of π (U) Γexc over a connected domain U ⊂ C , T −1 T and π(A) U = ∅. Then π (U) Γexc is the disjoint union of a finite number of open

connected sets Uν, each located in its own hyperplane. Assume that on each Uν there

is a holomorphic branch of the (multivalued) analytic function ln `. Define ωexc,` by the

formula ωexc,` = (ln `) ω, where ω is a rational n-form on Γexc, considered as a hypersurface

of degree d with d the number of hyperplanes in Γexc.

Lemma 7.8. For a different choice of the polynomial `, say `1, the restriction of ` and

`1 to each hyperplane l0 ⊂ Γexc coincide up to multiplication by a constant, depending on

the hyperplane `0. In addition, the trace of ωexc,` under the mapping π is a rational form

P`/Q`dz1 ∧ ... ∧ dzn on U if and only if traceπω = 0. Chapter 7. Applications of Main theorems and our methods 93

Thus ωexc,`1 = ωexc,` + ω with ω a rational n-form on Γexc. Whenever only the class of

ωexc,` modulo rational n-forms matters, we denote ωexc,` by ωexc.

Proof of Lemma 7.8. We start with the first assertion. Introduce an affine system of coordinates in Cn+1 in which the (n − 1)-dimensional plane A passes through the origin.

n+1 ∗ It allows us to think of ` and `1 as the elements of (C ) . Let `0 ⊂ Γexc be a hyperplane; we will think of `0 and A as an n-dimensional and (n − 1)-dimensional vector spaces, respectively.

Since ` and `1 vanish on A, the restriction of `1 to `0, as well as the restriction of ` to

∗ `0, is a non-zero element of (`0/A) . The vector space `0/A is one-dimensional, therefore,

∗ the vector space (`0/A) of linear functionals on `0/A is one-dimensional as well, and

thus `|`0 and `1|`0 coincide up to multiplication by a non-zero complex number. To show the rationality of the trace form and the converse statement, we introduce a system of affine coordinates (w, z1, . . . , zn) in which π is the projection along the w-axis and the plane A is given by the equations w = z1 = 0. If ωexc = (ln z1)ω, where ω is a

rational n-form on Γexc, then traceπωexc = (ln z1)traceπω.

Now ωexc,` is equal to (ln z1)ω up to a rational n-form. Since the trace of a rational form is rational (as follows from Theorem 3.1 for n = 1, and together with Proposi-

tions 2.20, 2.21 for an arbitrary n), we conclude that if traceπω = 0, then the trace of

ωexc,` under the mapping π is rational, and, conversely, if the trace of ωexc,` under the

mapping π is rational, then traceπω = 0. This completes the proof.

n+1 n Let γ1, . . . , γd be n-dimensional surfaces in C = C × C that are the graphs of

n holomorphic vector functions gi : U 7→ C defined in a connected domain U of C . Let π

be the projection of Cn+1 onto the first multiplier. The line π−1(0) is called vertical. Assume that, for any line L that is sufficiently close to the vertical one, there is a

n connected domain UL in C that differs fairly little from U so that the parallel projection Chapter 7. Applications of Main theorems and our methods 94

n −1 πL of γ1, . . . , γd onto C along L defines a d-sheeted covering of the preimage πL (UL) ⊆ S n+1 γi onto UL. Denote the standard holomorphic volume form on C by Ω. Finally, if a rational function F is the quotient of two relatively prime polynomials P and Q in n complex variables, then the maximum of the degrees of P and Q is called the degree of the rational function F .

Theorem 7.9. Let ω1, . . . , ωd be holomorphic forms of the highest degree on γ1, . . . , γd that do not vanish identically. Suppose that for any line L that is sufficiently close

to the vertical one, the trace of ω1, . . . , ωd under the mapping πL is a rational form

PL/QLdz1 ∧ ... ∧ dzn on UL and the degrees of the rational functions PL/QL are bounded by a single constant N then

1. In Cn+1, there is a (singular) n-dimensional algebraic variety Γ of degree d that

contains γ1, . . . , γd.

2. On Γ T π−1(U), there is a holomorphic form ω so that (Γ T π−1(U), ω) is a finite

T −1 sum of (Γν π (U), ων) where each (Γν, ων) is either an algebraic hypersurface

n+1 Γν ⊂ C together with a rational form on it or the union of hyperplanes Γexc

together with a form ωexc; the restriction of ω to each surface γk coincide with ωk.

The following two lemmas are direct consequences of Proposition 7.5 when n = 1. We

will need them for the proof of Theorem 7.9.

Lemma 7.10. Under the notations in Proposition 7.5 for n = 1, assume additionally

that γ1, . . . , γd belong to an algebraic curve Γ of degree d. Then there is a finite number of

rays I1,...,IN in C with any two of them either having empty intersection or one being S a subset of another so that U0 ⊂ Ω = C\( Ik) and

1. The functions gi : U 7→ C, which define the curves γi, allow for holomorphic S continuation to Ω ⊃ U0 resulting in the curves γi|Ω with the projection π : γi|Ω 7→ Ω d-sheeted over Ω. Chapter 7. Applications of Main theorems and our methods 95

2. The function R, which is holomorphic on U0×C, extends to a meromorphic function

RΩ on Ω × C.

3. The functions ρi also extends to meromorphic functions ρi|Ω on the curves γi|Ω and

the restriction of RΩ to γi|Ω is ρi|Ω.

In addition, the finite set of the initial points of the rays contains the projections of all

the singular points of Γ.

For a point p ∈ C2 = C × C we denote the vertical line passing through p ∈ C2 by

`p. Under the notations in Proposition 7.5 for n = 1, the function R has the form

R = (S0 + S1 ln(z − a1) + ...Sm ln(z − am))/Q. We suppose that m ≥ 1. We shall say

that R has a logarithmic singularity at ak if the polynomial Sk is not identically zero on

the union of curves γ1, . . . , γd.

Lemma 7.11. Under the notations in Proposition 7.5 for n = 1, assume additionally that

γ1, . . . , γd belong to an algebraic curve Γ of degree d, and suppose the mapping π :Γ 7→ C, induced by the projection of C2 = C × C onto the first multiplier, satisfies the properties 1, 2 below:

T 1. If p ∈ Γ is a singular point then the set of the intersection points `p (Γ\{p}) is

non-empty and consists of non-singular points of Γ at which `p transversely meets the curve.

2. If p ∈ Γ is a singular point and Γ0 is an irreducible component of Γ that is distinct

from a line with p ∈ Γ0, then the set of the intersection points of `p with Γ0\{p} is non-empty.

Suppose that the function R has logarithmic singularities at the projections of the singular points of Γ only, and, given a singular point p ∈ Γ, at each point of the transversal inter- T section `p (Γ\{p}) the function ρk|Ω with the appropriate index k admits meromorphic continuation into a small neighborhood of the intersection point. Chapter 7. Applications of Main theorems and our methods 96

Then:

1. If Γ0 is an irreducible component of Γ that is distinct from a line, then the function

T −1 R from Proposition 7.5 restricted to Γ0 π (U0) is rational.

2. For each ak ∈ C there is a finite union of lines Γexc ⊂ Γ passing through a point

that projects, under the mapping π, into the point ak ∈ C so that the restriction of T −1 R from Proposition 7.5 to Γexc π (U0) is (S0 + Sk ln(z − ak))/Q with S0,Sk,Q polynomials on 2. In addition, the trace (S /Q)dz = 0. C π|Γexc k

To show the existence of a projection as in Lemma 7.11, we will need the following geometric

Lemma 7.12. Suppose that Γ ⊂ C2 is an algebraic curve without multiple irreducible components and `0 is a line passing through a singular point p ∈ Γ. Then either Γ is a finite union of lines passing though one point or for any line ` passing through the

2 T point p ∈ C that is sufficiently close to `0 the set of the intersection points ` (Γ\{p}) is non-empty and consists of non-singular points of Γ at which ` transversely meets the curve.

Proof of Theorem 7.9. Step 1: Theorem 7.9 (1).

Let a coordinate w be fixed on the line C, and Ω0 = dz1 ∧...∧dzn be the holomorphic volume form on Cn. By computing derivatives as in Step 1 in the proof of Theorem 5.1,

k we find that, for any positive integer k, all the kth order derivatives of the form traceπw ω

n along constant vector fields in C are rational Pk/Qkdz1 ∧ ... ∧ dzn. Now, let ρi be the function defined from the identity traceπωi = ρiΩ0 and wi be the restriction of w to γi. In particular, it follows that (4d − 1)th order derivatives along any constant vector field

n k k n in C of the functions sk = ρ1w1 +...+ρdwd with 0 ≤ k ≤ 4d−1 are rational on U ⊂ C . By Proposition 7.5 (applied to a small neighborhood of a point at which the functions

n+1 ρk does not vanish) and analyticity of γ1, . . . , γd, there is a polynomial P in C whose Chapter 7. Applications of Main theorems and our methods 97

zero locus in U × C is the union of all the surfaces γν. To show that there is an algebraic

hypersurface of degree precisely d that contains the surfaces γ1, . . . , γd, we employ the same arguments as in the last paragraph of Step 1 in the proof of Theorem 5.1.

Step 2: Theorem 7.9(2), n = 1.

Suppose that the curves belongs to an algebraic curve Γ of degree d. By Lemma 7.12,

we can choose a projection of Γ along a line onto the complex line with the properties

1, 2 in Lemma 7.11. Thus without loss of generality we may assume that the vertical

projection π :Γ 7→ C satisfies properties 1, 2 in Lemma 7.11. Let ρ1|Ω, . . . , ρd|Ω be

the functions on the curves γk|Ω constructed in Lemma 7.10. Denote the corresponding forms ρk|Ωdz by ωk|Ω. By analyticity, the trace of ω1|Ω, . . . , ωd|Ω under the projection πL is rational, if a line L is sufficiently close to the vertical.

Suppose that p ∈ Γ is a singular point and the line `p transversely meets Γ at a point

p1 ∈ Γ. By choosing a projection that is slightly different from the vertical, we conclude

that the form ωk|Ω with the appropriate index k has meromorphic continuation to a neighborhood of the point p1 ∈ Γ. Thus the corresponding function ρk|Ω has meromorphic continuation to a neighborhood of p1 ∈ Γ.

Now suppose that a ∈ C is the initial point of some ray Ik constructed in Lemma 7.10 and a ∈ C is not the projection of a singular point of Γ, in other words, the preimage

−1 π (a) = {m1p1, . . . , msps} consists of a finite number of non-singular points pν ∈ Γ together with multiplicities mν ∈ N. Again, by choosing a projection that is slightly different from the vertical, we conclude that near each p ∈ Γ the forms ω | , . . . ω | k k1 Ω km1 Ω with the appropriate indices k1, . . . , km1 coincide with a meromorphic form in a neigh- borhood of the point pk ∈ Γ. Thus the function RΩ|Γ is univalued near the points p1, . . . , ps. We conclude that the function R may have logarithmic singularities only at the projections of singular points of Γ.

Now if the function R, constructed locally, from Proposition 7.5 has no logarithmic singularities at all, then R is rational and we are done. Otherwise, Lemma 7.11 implies Chapter 7. Applications of Main theorems and our methods 98

that:

1. If Γ0 is an irreducible component of Γ that is distinct from a line, then there is

T −1 a rational 1-form Ω that restricted to γk|U0 ⊂ Γ0 π (U0) coincide with ωk. By

analyticity, the restriction of Ω to the whole curve γk coincide with ωk.

2. For each ak ∈ C there is a finite union of lines Γexc ⊂ Γ passing through a point

2 Ak ∈ C that projects, under the mapping π, into the point ak ∈ C, and a 1-form

Ω = Ω1 + ln(z − ak)Ω2 with Ω1 and Ω2 rational on Γexc so that the restriction of Ω

T −1 to γk|U0 ⊂ Γexc π (U0) coincide with ωk. In addition, the trace of Ω2 under the

mapping π|Γexc is identically zero.

T −1 We conclude that on each γν ⊂ Γexc π (U) there is a holomorphic branch of ln(z −ak),

T −1 and by analyticity, the restriction of Ω to the whole curve γν ⊂ Γexc π (U) coincide

with ων. To complete the proof we show that trace Ω = 0 for a projection π that is πL|Γexc 2 L

sufficiently close the vertical and with Ω2 as above. Indeed, if the projection πL takes the points A ,...,A into pairwise distinct points, then trace ln(z −a )Ω is univalued 1 m πL|Γexc k 2

near π(Ak), and thus it is a rational 1-form on the complex line. By Proposition 7.8, trace Ω = 0. We conclude that Ω is a generalized holomorphic form on Γ . πL|Γexc 2 2 exc Step 3: Theorem 7.9(2), n ≥ 2.

The case n ≥ 2 we reduce to the case n = 1 by choosing appropriate plane sections. Let

Γ be an algebraic hypersurface of degree d that contains γ1, . . . , γd. Suppose that Γ0 is an irreducible component that is distinct from a hyperplane. We show that the function

R from Proposition 7.5 is rational on Γ0. Indeed, suppose for contradiction that, say

S1, does not vanish identically on Γ0, where R = (S0 + S1 ln P1 + ...Sm ln Pm)/Q with

n+1 Pk,Sk,Q polynomials on C .

Let Σ ⊂ Cn be an algebraic hypersurface that contains both the hypersurface of criti- cal values of the projection π :Γ 7→ Cn and the hypersurfaces defined by the polynomials Chapter 7. Applications of Main theorems and our methods 99

n P1,...,Pm. Choose a point p ∈ U and a line l ∈ C with p ∈ l so that 1) the line l transversely meets Σ at degΣ points; 2) the second derivative of each function gi, which

graph is the surface γi, in a direction of the line l evaluated at the point p ∈ U is not

−1 zero; 3) the polynomial S1 does not vanish at the preimages π (p) located on Γ0; 4)

−1 each ρk does not vanish at the point of π (p) that belongs to the surface γk.

Let L be the 2-dimensional plane that is the preimage of l ⊂ Cn under the projection

of Cn+1 = Cn × C onto the first multiplier. Then the first condition above ensures that Γ T L is an algebraic curve of degree d located in the plane L. The second condition T ensures that each irreducible component of curve Γ0 L is distinct from a line. Now, if a line l ⊂ Cn satisfies only the first condition above and is defined by (n − 1) linear poly- ω T nomials `1, . . . , `n−1, then I claim that the form ωL = ∗ ∗ on Γ L satisfies dπ `1∧...∧dπ `n−1 the following two properties:

1. The trace of ωL under a parallel projection π` along a line ` that is sufficiently close

to the vertical and is located in the plane L onto the line l is rational P`/Q`dw, where w is an affine coordinate on l.

2. Up to multiplication by a non-zero complex constant, ωL = R|Γ T Ldw.

traceπ` ω The identity (3.2) on the page 56 implies that traceπ ωL = ∗ ∗ , and this ` dπ `1∧...∧dπ `n−1 explains the first property. To show the second property, note that dw does not vanish

on the line l, so dw ∧d`1 ∧...∧d`n−1 = Cdz1 ∧...∧dzn with C ∈ C a non-zero constant.

We conclude that, up to multiplication by a non-zero constant, ωL = R|Γ T Ldw. Now we return to the proof. Suppose that l is a line that satisfies conditions 1), 2), 3), 4)

above. By Theorem 7.9(2) case n = 1, the form ωL|Γ0 is rational. We have a contradiction

with Lemma 7.13 since S1|Γ0 is not identically zero. Thus it is sufficient to prove Step 4: Theorem 7.9(2) with n ≥ 2 and Γ a finite union of hyperplanes.

n+1 S So suppose that Γ ⊂ C is a finite union of hyperplanes Lν. Without loss of generality

we assume that the projection π :Γ 7→ Cn sends geometrically distinct intersections Chapter 7. Applications of Main theorems and our methods 100

T n Li Lj of various dimensions into distinct planes of the same dimension in C . By choosing appropriate plane sections and using the properties 1, 2 of the forms

ωL above, we show that 1) if a hyperplane, say L1, has no (n − 2)-dimensional inter- section with other hyperplanes, then the function R from Proposition 7.5 restricted to

T −1 n L π (U0) is rational; 2) the polynomials P1,...,Pm ∈ C in the representation of the function R are linear, and define the projections of (n − 2)-dimensional mutual intersec- tions of the hyperplanes in Γ; 3) for each polynomial Pk, there is a certain Γexc,k passing

n+1 through an (n − 2)-dimensional plane Ak ⊂ C which image π(Ak) is defined by the

n+1 polynomial Pk; 4) there are rational n-forms Ω1 and Ω2 on C so that the restriction T −1 of Ω1 + ln(Pk)Ω2 to the surfaces γν ⊂ Γexc,k π (U0) coincide with ων.

T −1 We conclude that on each γν ⊂ Γexc,k π (U) there is a holomorphic branch of

T −1 ln(Pk), and by analyticity, the restriction of Ω to the whole curve γν ⊂ Γexc,k π (U) coincide with ωk To complete the proof we show that trace Ω = 0 for a projection π that πL|Γexc 2 L

is sufficiently close the vertical and with Ω2 as above. Indeed, if the projection πL

takes the (n − 2)-dimensional planes A1,...,Am into pairwise distinct hyperplanes in n, then trace ln(P )Ω is a rational n-form on n, and thus, by Proposition 7.8, C πL|Γexc k 2 C trace Ω = 0. We conclude that Ω is a generalized holomorphic form on Γ . πL|Γexc 2 2 exc

Proof of Lemma 7.10. Let A ⊂ C be the union of the finite set of critical values of the

projection π :Γ 7→ C and the set of points a1, . . . , am, the poles of all the Nth order

derivatives of s0, . . . , s4d−1. In particular, A contains the image of singular points of the curve Γ.

To obtain the rays Ik, connect each point of A with the north pole N of the Riemann sphere CP 1 via the shortest (any of the two for the south pole of the Riemann sphere) arc on the great circle, and then use the stereographic projection of the Riemann sphere onto the complex line C. The image of such an arc is a ray. Clearly, different points Chapter 7. Applications of Main theorems and our methods 101

of the set A gives rise to either two rays with empty intersection or two rays one inside

another.

Now the domain Ω is simply connected and any point of Ω has d geometrically distinct

−1 preimages in Γ. Thus π (Ω) is a disjoint union of d curves γ1|Ω, . . . , γd|Ω that are the graphs of holomorphic functions gk|ω :Ω 7→ C that coincide with the functions gi : U 7→ C on U ⊂ Ω. By choosing holomorphic branches of ln(z−aj) on a simply-connected domain

Ω ⊂ C, we construct the analytic continuation RΩ and ρk|Ω of the functions R and ρk, respectively, and thus complete the proof.

Below is a well-known fact which we need in the proof of Theorem 7.9(2). Suppose that

π :Γ 7→ CP 1 is a proper finite holomorphic, in the sense of algebraic varieties, mapping between a projective curve Γ, possibly singular but without multiple components, and

1 the Riemann sphere. Let z1, . . . zm be points in the affine chart C ⊂ CP with a fixed

coordinate z, and U ⊂ C\{z1, . . . , zm} be a connected and simply-connected domain. On

U, fix holomorphic branches of the functions ln(z − zk).

Lemma 7.13. Suppose that R0,...,Rm are rational functions on Γ. Then the function

∗ ∗ −1 R = R0 + R1π ln(z − z1) + ... + Rmπ ln(z − zm) on π (U) ⊂ Γ coincide with a rational function on Γ if and only if R1,...,Rm are identically zero.

Proof of Lemma 7.11. Let a1, . . . , am ∈ C be the images of singular points of Γ under the projection π :Γ 7→ C. Then the function R has the form R0 +R1 ln(z−a1)+...Rm ln(z− am) with Rk rational on Γ.

Let Γ0 be an irreducible component of Γ that is distinct from a line. Fix some ak.

Properties 1, 2 ensure that the line `ak meets Γ0 transversely at a non-singular point a ∈ Γ0. Near a ∈ Γ0 the function (z −ak) is a local coordinate on Γ0. By the assumption, the function RΩ restricted to a certain branch is univalued near a ∈ Γ0. Since Γ0 is

an irreducible component, we conclude that Rk|Γ0 is identically equal to zero. These Chapter 7. Applications of Main theorems and our methods 102

arguments are valid for each ak, thus the restriction of R to Γ0 coincide with a rational function.

Since the trace of a rational 1-form is rational, we may assume that Γ is a finite

union of lines on the plane. We will prove Lemma 7.11(2) by induction on the number

of logarithmic singularities that the function R = R0 + R1 ln(z − a1) + ...Rm ln(z − am),

which is holomorphic on U0 × C, has. Fix a singular point of Γ and consider all the lines

in Γ that pass through that singular point. They form a certain Γexc. Let aj ∈ C be the image of the singular point under the projection π :Γ 7→ C.

Now consider new functionsρ ˜k = ρk − Rj|γk ln(z − aj) on the curves γ1, . . . , γd (re- stricted to U0 ⊂ C). I claim that:

1. The restriction of Rj to any line not in Γexc is identically zero, and thusρ ˜k = ρk for the appropriate indices k.

2. If N is as Proposition 7.5, then the Nth order derivatives of the functionss ˜k =

k k ρ˜1w1 + ... +ρ ˜dwd (0 ≤ k ≤ 4d − 1 ) with respect to z are rational on C.

We first show how to complete the proof of Lemma 7.11(2). After removing some lines from Γexc at which the new functions are identically zero, the application of the induction assumption (together with Lemma 7.13 if aj is no longer a singular point of the new curve) completes the proof.

We start with the first assertion. If ` ⊂ Γ\Γexc is a line, then `aj meets ` transversely at some point near which the function RΩ is univalued. Thus Rj restricted to the line ` is identically zero.

P k To show the second claim, note thats ˜k − sk = Rj|γν ln(z − aj)w , with the sum- ν mation over all the lines γν in Γexc. By subtracting the logarithm in the definition of

ρ˜, we ensured that the natural analytic continuations ˜k|Ω of a functions ˜k to the do- main Ω is univalued near aj ∈ C. Thus the Nth order derivatives of the functions P k P k Rj|γν w ln(z − aj) is univalued near aj ∈ C. Now the function Rj|γν w ln(z − aj) ν ν Chapter 7. Applications of Main theorems and our methods 103 has the form G ln(z − aj) with G a rational on C. We conclude that the Nth order P k derivatives of the functions Rj|γν ln(z − aj)w with respect to z are, in fact, rational. ν k k Thus the Nth order derivatives of the functionss ˜k =ρ ˜1w1 + ... +ρ ˜dwd (0 ≤ k ≤ 4d − 1) are rational on C.

Proof of Lemma 7.12. Let L ⊂ C2 be a line that does not belong to Γ. Consider the projection π :Γ\{p} 7→ L¯ from the point p ∈ C2 to the projective closure of L. I claim that if a ∈ L¯ is a non-critical value of the mapping π :Γ\{p} 7→ L¯, that is dπ does not vanish at all the points of π−1(a), then the line connecting p ∈ Γ with the point a ∈ L¯ transversely meets Γ at all the preimages π−1(a); all the preimages are non-singular points on the curve. Indeed, in an affine chart of L¯ containing the point a, the mapping π is given by the following formula: π = `1/`2, where `1, `2 are linear

−1 polynomials in two complex variables, `1(p) = `2(p) = 0, and `2(q) 6= 0. Since q ∈ π (a),

`1 `1−a`2 d(`1−a`2) we have `1(q) − a`2(q) = 0, and thus d( ) = d( ) = 2 at the point q ∈ Γ. `2 `2 `2 We conclude that 1) the point q ∈ Γ is non-singular; 2) the line defined by the equation

`1 − a`2 = 0 transversely meets Γ at q ∈ Γ.

Now, if Γν is an irreducible component of Γ, then its image π(Γν) is either the whole ¯ L (maybe without one point if p ∈ Γν) or just one point. In the latter case Γν is a line passing through the point p ∈ C2. So if Γ is not a finite union of lines passing through the point p ∈ C2, then by choosing a non-critical value of the mapping π :Γ\{p} 7→ L¯ T ¯ near the point `0 L we complete the proof. Chapter 8

Cotes’ theorem an its converse

8.1 Cotes’ theorem

Suppose that Γ ⊂ R2 is an algebraic curve of degree d, and L is a family of parallel lines such that each line ` ∈ L meets the curve Γ at d points, counted with multiplicities. T Denote by M` the center of masses of the intersection points ` Γ. Under these notations the following theorem is valid

Theorem 8.1 (Cotes’ theorem). The centers of masses M`, where ` runs over all the lines in the family L, belong to one line.

Proof. We choose affine coordinates (x, y) on the plane so that L belongs to the family of vertical lines x = const. We rewrite the polynomial P that defines Γ in the following

d d−1 way P (x, y) = y + p1(x)y + ... + pd(x) with coefficients pj ∈ R[x] polynomials of the degree at most j. Let y1(c), . . . , yd(c) be the y-coordinates of the intersection of the curve

Γ with a vertical line x = c (c ∈ R). Then, by Vieta’s formula, y1(c)+...+yd(c) = p1(c).

It means that the y-coordinate of the center of masses My=c linearly depends on its x-coordinate c. Thus points M` belong to one line.

Remark 8.1. Near a generic c ∈ R functions y1(c), . . . , yd(c) are smooth. Differentiating

00 00 the identity y1(c) + ... + yd(c) = p1(c) twice we get y1 (c) + ... + yd (c) = 0. The

104 Chapter 8. Cotes’ theorem an its converse 105

00 kj straightforward computation shows that yj (c) = sin3 θ , where kj is the curvature of Γ at the point Pj(c) = (c, yj(c)), and θj is the angle that the tangent to Γ at the point Pj(c) makes with the y-axis. Consequently, Cotes’ theorem has the following nice differential- geometric form.

Given a line l that meets Γ at d distinct points P1,...,Pd then

k1 kd 3 + ... + 3 = 0, sin θ1 sin θd

where kj is the curvature of Γ at the point Pj and θj is the angle that the tangent to Γ at

the point Pj makes with the line l.

The latter identity is called Reiss relation.

Now we proceed with a multidimensional generalization of Cotes’ theorem. Assume

that Γ ⊂ Rn is a k-dimensional algebraic variety of degree d, and L is a family of parallel (n − k)-dimensional planes such that each ` ∈ L meets the variety Γ at d points, counted with multiplicities.

Theorem 8.2 (Multidimensional Cotes’ theorem). The centers of masses M`, where ` runs over all the (n − k)-dimensional planes in the family L, belong to a single k- dimensional plane.

Proof. The proof consists of two steps. In the first step we show the theorem when Γ is a hypersurface in Rn. In the second step we treat the general case by reducing it to the hypersurface case.

n Step 1. If k = n − 1, we choose affine coordinates (x, y) = (x1, . . . xn−1, y) in R =

n−1 R × R so that L belongs to the family of vertical lines x1 = c1, . . . , xn−1 = cn−1. After that the proof follows the same lines as the proof of the previous theorem.

Step 2. In the general case we choose affine coordinates (x, y) = (x1, . . . xk, y1, . . . , yn−k)

n k n−k in R = R × R so that L is the family of vertical planes x1 = c1, . . . , xk = ck, and Chapter 8. Cotes’ theorem an its converse 106

yi(Pj) 6= yi(Ps) for all 1 ≤ i ≤ n − k and 1 ≤ s, j ≤ k, where P1,...,Pd are the intersec-

tion points of the vertical plane x1 = ... = xk = 0 and the variety Γ. The latter condition ensures that, for any 1 ≤ j ≤ n − k, the projection of Γ into the affine subspace with

coordinates x1, . . . , xk, yj is the degree d hypersurface. Finally, we notice that, by the

previous step, yj(M`) linearly depends on x1, . . . , xk. Thus M` belong to a k-dimensional plane as the line ` varies in the family L.

8.2 The converse of Cotes’ theorem

We follow the notations of the converse of Abel’s theorem. Let γ1, . . . , γd be n-dimensional

n+k n k k surfaces in R = R × R that are the graphs of vector functions gi : U 7→ R defined

in a connected domain U of Rn. Let π be the projection of Rn+k onto the first multiplier. The k-dimensional plane π−1(0) is called vertical.

Assume that, for any k-dimensional plane L that is sufficiently close to the vertical

n one, there is a connected domain UL in R that differs fairly little from U so that the

n parallel projection πL of γ1, . . . , γd onto R along L defines a d-sheeted covering of the −1 S preimage πL (UL) ⊆ γi onto UL.

n+k The preimage of UL under πL is a pencil of parallel k-dimensional planes in R , −1 which is denoted by πL .

Theorem 8.3 (Converse of Cotes’ theorem). Assume that, for any k-dimensional plane

L that is sufficiently close to the vertical one, the centers of masses of the d intersection

points of the planes from with the surfaces γ1, . . . , γd lie in a single n-dimensional plane (depending on the pencil of parallel hyperplanes).

Then there is a (singular) n-dimensional algebraic variety γ of degree d in Rn+k that

contains γ1, . . . , γd.

We will need the following well-known fact and its geometric Corollary 8.5. For the proof Chapter 8. Cotes’ theorem an its converse 107

see Proposition 2.21.

n Proposition 8.4. Let v1, . . . , vn be linearly independent vectors in R . Suppose that f

is a complex-valued function on a connected domain U ⊂ RN such that for any line l

parallel to one of the directions vk the restriction f|l T U is a polynomial of degree at most

nf , where nf does not depend on the line l. Then f coincide with a polynomial in n

variables of degree at most nf .

N+1 N Corollary 8.5. Let γ1, . . . , γd ⊂ R = R × R be graphs of functions defined on a

connected domain U ⊂ RN , and π be the projection of RN+1 onto the first multiplier.

n N Let v1, . . . , vn be linearly independent vectors in R . Assume that for any line l ⊂ R

−1 parallel to one of the directions vk the preimage π (l) is an algebraic curve of degree d.

N+1 Then there is a degree d algebraic hypersurface in R that contains γ1, . . . , γd.

Proof of Theorem 8.3. Applying the projection onto a lower dimension space (Proposi-

tion 2.16), we can assume that the surfaces γ1, . . . , γd are the graphs of functions defined

on a connected domain U of Rn. By corollary 8.5 it is sufficient to prove the converse Cotes’ theorem when n = k = 1.

Now we consider the case when n = k = 1. Notice that for any graph γk, up to proportionality, there are only countably many linear functionals l ∈ (R2)∗ so that the zero set of dl|γk is not nowhere dense on γk. Indeed, if the set of zeros of dl|γk is dense near a point P of, say, γ1, then the linear functional l is constant in a sufficiently small neighborhood UP of P . Now in each such U we can choose a point with a rational projection of R × R onto the first multiplier.

Choose a linear functional l so that dl vanishes on nowhere dense subsets of γ1, . . . , γd.

The trace of dl under πL is a multiple of the standard volume form dx. Application of the converse Abel’s theorem completes the proof.

Remark 8.2. In the conditions of the converse of Cotes’ theorem, it is sufficient to require Chapter 8. Cotes’ theorem an its converse 108

1 that γ1, . . . , γd be the graphs of C vector functions.

The next is a straightforward corollary from the converse of Cotes’ theorem

Corollary 8.6. Suppose that U ⊂ R2 is a convex domain bounded by a C2-smooth Jordan curve. Assume that for any family of parallel lines the middle points of the intersection segments with the domain U belong to one line (depending on the family of parallel lines).

Then U is the interior of an ellipse. Chapter 9

Hypersurfaces of double translation

9.1 Surfaces of double translation

Let S be a smooth surface in R3. The properties of S that we discuss below are invariant under translations in the ambient space, so in this section we will always assume that S contains the origin.

Definition 1. The surface S is called a translation surface if there exists two smooth

3 parameterized curves γi : ∆i 7→ R (i = 1, 2) defined on intervals ∆i ⊂ R of the real line so that every point x ∈ S can be uniquely written in the form x = γ1(t1) + γ2(t2), where

(t1, t2) ∈ ∆1 × ∆2. Additionally, we assume that the curves γi(∆i) contain the origin in

3 R . We write S = γ1 + γ2.

3 Below when there is no ambiguity we will refer to the image γi(∆i) ⊂ R as to the curve

γi. The mechanical interpretation of a translation surface S is the following. Fix one

3 curve, say γ1 ⊂ R , and move the other one γ2 rigidly (parallel to itself) along the fixed one. The locus of the points that is swept out by such motion is the surface S. We could have obtained the surface S by fixing the curve γ2 and moving γ1 along it as well. Thus there are two families of parallel curves on the surface S.

Example 2 (Translation surfaces). Plane, cylinder. Notice that these surfaces have in-

109 Chapter 9. Hypersurfaces of double translation 110

finitely many translation surface structures, that is the choices of curves γ1 and γ2 from the definition 1.

The term ”distinct” in the next definition we will clarify further.

Definition 2. The surface S is called a surface of double translation if there exists two

”distinct” representation of S as a translation surface S = γ1 + γ2 and S = γ3 + γ4. Each

γi is defined on its own interval ∆i.

Example 3 (Double translation surfaces). Let Γ be degree four algebraic curve on the

2 real plane R and ` a line that transversely meets Γ at four points B1,B2,B3,B4. Let

Γ1, Γ2, Γ3, Γ4 ⊂ Γ be the curves, with Bi ∈ Γi, swept out by all the lines in a connected

2 ∗ neighborhood U` of ` in the space of lines (RP ) . Suppose that

1. The curves Γi are disjoint and belong to the non-singular part of Γ.

2. Each Γi is a parameterized curve Γi : ∆i 7→ Γ defined on its own interval ∆i of the

real line. Each mapping Γi is injective with dΓi not vanishing on ∆i.

Let P ∈ R[x, y] be a degree four polynomial that defines the curve. Choose a real- Qdx∧dy valued basis ω1, ω2, ω3 of the 3-dimensional space { dP |degQ ≤ 1} of generalized 3 holomorphic forms on Γ. Let γ1, γ2, γ3, γ4 ⊂ R be the curves parameterized by points of

Γ1, Γ2, Γ3, Γ4 ⊂ Γ respectively in the following way:

 A A A   A A A  Z 1 Z 1 Z 1 Z 1 Z 1 Z 1 γ1(A1) =  ω1, ω2, ω3 , γ2(A2) =  ω1, ω2, ω3 ,

B1 B1 B1 B1 B1 B1

 A A A   A A A  Z 1 Z 1 Z 1 Z 1 Z 1 Z 1 γ3(A3) = −  ω1, ω2, ω3 , γ4(A4) = −  ω1, ω2, ω3 ,

B1 B1 B1 B1 B1 B1 where Ak ∈ Γk. Different choice of the basis ω1, ω2, ω3 leads to a linear transformation

3 of the curves γ1, γ2, γ3, γ4 in R . Suppose that both γ1 + γ2 and γ3 + γ4 are smooth translation surfaces in R3. Chapter 9. Hypersurfaces of double translation 111

It follows from the Abel’s theorem that γ1(A1) + γ2(A2) = γ3(A3) + γ4(A4) for points

A1,A2,A3,A4 on Γ that belong to one line l ∈ U`. Indeed, due to connectivity of the domain U` we may assume that the line l is sufficiently close to `. Consider the

1 T projection π :Γ 7→ RP from the intersection point ` l (it could be at infinity) to a line that is transversal to both ` and l. Let A and B be the images of all the points

Ak and Bk respectively. According to Abel’s theorem traceπω = 0 for ω a generalized A A A A A R R1 R2 R3 R4 holomorphic form. In particular, traceπω = ω + ω + ω + ω is equal to zero. B B1 B2 B3 B4 Thus γ1(A1) + γ2(A2) = γ3(A3) + γ4(A4). We will show that the two representations of S as a translation surface are “distinct”, in the sense of definition below, in the proof

Theorem 9.2.

We finish this example by showing that

Proposition 9.1. For U` that is sufficiently small, both γ1 + γ2 and γ3 + γ4 are smooth translation surfaces in R3.

Proof of Proposition 9.1. Choose a line l that is transversal to `. Near the point Bν the curve Γ is the graph of a function defined on some interval on the line l.

If U` is sufficiently small, then, by the implicit function theorem, Γ1, Γ2, Γ3, Γ4 ⊂ Γ are disjoint and lie on the graphs of the functions as above. There is a natural diffeomorphism

2 between U` and Γ1 ×Γ2 that sends a point in U`, which is a line on R , to the intersection ∼ ∼ points with Γ1 and Γ2. Thus U` = Γ1 ×Γ2, and, similarly, U` = Γ3 ×Γ4. In particular, Γν is connected, and we conclude that Γν is the graph of a function defined on an interval of the line l.

3 Now we show that both γ1 +γ2 and γ3 +γ4 are smooth translation surfaces in R if U` is sufficiently small. I claim that the differential of the map sending a point (A1,A2) ∈

Γ1 × Γ2 to γ1(A1) + γ2(A2) has the maximal rank, and thus S = γ1 + γ2 is a smooth translation surface in a sufficiently small neighborhood of the origin in R3. Indeed, dx∧dy in the standard basis ω1 = ω, ω2 = xω, ω3 = yω, where ω = dP , the differential

dγk = (ω, xω, yω)|Γk (k = 1, 2). Thus, up to a non-zero constant, the differential of Chapter 9. Hypersurfaces of double translation 112

  1 x(A ) y(A )  1 1  the mapping in question is   . Similarly, the differential of the map 1 x(A2) y(A2)

sending a point (A3,A4) ∈ Γ3 × Γ4 to γ3(A3) + γ4(A4) has the maximal rank, and thus

γ3 + γ4 is a smooth translation surface in a sufficiently small neighborhood of the origin in R3.

The next is the key construction in the classification theorem of the surfaces of double

translation. It also provides a rigorous meaning for the “distinct” in the definition 4.

Lie’s construction. Let S ⊂ R3 be a surface with two representations as a translation

2 surface S = γ1 + γ2 and S = γ3 + γ4. Denote by Π = RP the infinity plane in

3 3 the projective closure of R . With each γi(∆i) ⊂ R we associate a parameterized curve T πγi : ∆i 7→ Π by the following rule: t ∈ ∆i is mapped into the intersectionγ ˙ i(t) Π of the

tangent lineγ ˙ i(t) with the infinity plane. Now, in the tangent plane TP S at a point P ∈ S

there are four distinguished directions: they are parallel toγ ˙ 1(t1), γ˙ 2(t2), γ˙ 3(t3), γ˙ 4(t4),

where P = γ1(t1) + γ2(t2) = γ3(t3) + γ4(t4). Thus the four points πγi (ti) belong to one

line — the intersection of TP S and the infinity plane. Two representations γ1+γ2 = γ3+γ4 of the surface S are called distinct if

1. Each mapping πγi : ∆i 7→ Π is an injection with dπγi not vanishing on ∆i, and the

curves πγi (∆i) are disjoint.

2 ∗ T 2. The map G : S 7→ (RP ) sending P ∈ S to the line TP S Π is a diffeomorphism onto the image G(S).

3. The line G(P ) ⊂ Π transversely meets each curve πγi (∆i) at πγi (ti).

It is a well-know fact that, locally, the second condition is equivalent to the non-degeneracy of the second quadratic form at each point of S. Proposition 9.4 is the next section pro- vides a local, quantitative description of the first and the third conditions. Chapter 9. Hypersurfaces of double translation 113

Now we are ready to state Lie’s classification theorem

Theorem 9.2. Let S be a C2-smooth surface in R3. Then S a surface of double transla- tion if and only if there exists an algebraic curve Γ ⊂ R2 of degree four, line ` transversely meeting Γ in four points, and a real-valued basis in the space of generalized holomorphic forms on Γ so that S is constructed as in the example 3.

Proof of Theorem 9.2. We use the notations of the example 3 and Lie’s construction.

Sufficiency. The surface S constructed in the example 3 is smooth and has two representations as a translation surface. We need to show that the two representations are distinct. As properties 1),2),3) are invariant under the linear action in R3, we may assume that S is constructed from the standard basis ω1 = ω, ω2 = xω, ω3 = yω. In this case Lie’s construction is the inverse of the construction in the example 3 in the following sense:

Ak = πγk (Ak) for a point Ak ∈ Γk (k = 1, 2, 3, 4). Indeed, if γk(t) = (x(t), y(t), z(t)) then

πγk (t) has homogeneous coordinates [x ˙(t) :y ˙(t) :z ˙(t)]. For γk defined by the formulas in

the example 3, we have dγk = (ω, xω, yω)|Γk . Thus πγk (Ak) has homogeneous coordinates

[1 : x(Ak): y(Ak)].

Necessity. Let S = γ1(∆1) + γ2(∆2) = γ3(∆3) + γ4(∆4) be a surface of double

3 3 ∗ translation in R . Consider any linear functional l ∈ (R ) . On each curve πγk (∆k) ⊂

2 RP in Lie’s construction define a function by the following rule: fk(t) = l(γk(t)) if k = 1, 2; fk(t) = −l(γk(t)) if k = 3, 4. If P = γ1(t1) + γ2(t2) = γ3(t3) + γ4(t4), then the

sum of values of the functions fk at the intersection points of G(P ) with curves πγk is equal to zero.

If the curves γk = γk(∆k) were real-analytic, we would choose coordinate functions

3 ∗ l1, l2, l3 ∈ (R ) so that each dlk does not vanish at one particular point on each curve.

Then it follows from the analyticity, that each dlk vanishes on nowhere dense subsets on

γ1, . . . , γ4. The application of Theorem 7.1 to each lk would complete the proof. Since the curves are just smooth, we need some extra work. In fact, the extra arguments in the multidimensional case are the same as for surfaces. That is why we refer to the end Chapter 9. Hypersurfaces of double translation 114

of the proof of Theorem 9.7, where one should substitute n = 2.

9.2 Multidimensional case

Now we proceed with to the multidimensional case. Following the spirit of this manuscript

we will describe the real case indicating, when necessary, modifications in the analytic

case. In fact, one of the applications – Torelli theorem requires the complex-analytic

counterpart of the theorem 9.9 below.

Let S be a smooth hypersurface in Rn+1 (or a smooth analytic hypersurface in Cn+1). The properties of S that we discuss below are invariant under translations in the ambient

space, so in this section we again assume that S contains the origin. Definitions 3, 4 are

straightforward generalizations of the concepts introduced in the previous section.

Definition 3. The hypersurface S is called a translation hypersurface if there exists n

n+1 smooth parameterized curves γi : ∆i 7→ R defined on intervals ∆i ⊂ R of the real line, so that every point x ∈ S can be uniquely written in the form x = γ1(t1) + ... + γn(tn),

n+1 where tν ∈ ∆ν. Additionally, we assume that the curves γi(∆i) contain the origin in R .

We write S = γ1 + ... + γn

Remark 9.1. In the analytic case intervals ∆i are replaced by connected and simply-

connected domains ∆i ⊂ C.

The term “distinct” in the next definition is explained in Lie’s construction.

Definition 4. The hypersurface S is called a hypersurface of double translation if there

exists two ”distinct” representation of S as a translation surface S = γ1 + ... + γn and

S = γn+1 + ... + γ2n. Each γi is defined on its own interval ∆i.

Replacing Π = RP 2 by hyperplane at infinity RP n in the projective closure of Rn+1, Lie’s construction, with obvious modifications, extends to the multidimensional case. Chapter 9. Hypersurfaces of double translation 115

We denote by L(S) the disjoint union of of 2n curves πγi (∆i) in Π. Two representations

γ1 + ... + γn = γn+1 + ... + γ2n of the hypersurface S are called distinct if

1. Each mapping πγi : ∆i 7→ Π is an injection with dπγi not vanishing on ∆i; the

curves πγi (∆i) are disjoint.

n ∗ T 2. The map G : S 7→ (RP ) sending P ∈ S to the hyperplane TP S Π is a diffeo- morphism onto the image G(S).

3. The hyperplane G(P ) ⊂ Π transversely meets each curve πγi (∆i) at πγi (ti), where

P = γ1(t1) + ... + γn(tn) = γn+1(tn+1) + ... + γ2n(t2n).

We shall say that S is a hypersurface of double translation near its point a ∈ S if S shifted by the vector a is a hypersurface of double translation. The next proposition provides a criterion for a hypersurface with two translation structures to, locally, be a hypersurface of double translation. The proof is, essentially, the use of the implicit function theorem.

Proposition 9.3. Let S ⊂ Rn+1 be a smooth hypersurface having two representations as a translation hypersurface S = γ1 +...+γn and S = γn+1 +...+γ2n. Suppose that a ∈ S and a = γ1(t1) + ... + γn(tn) = γn+1(tn+1) + ... + γ2n(t2n). Then there is a neighborhood of the point a in which S is a double translation hypersurface if and only if

1. The points πγi (ti) are pair-wise distinct and dπγi does not vanish at ti ∈ ∆i.

n ∗ T 2. The map G : S 7→ (RP ) sending P ∈ S to the hyperplane TP S Π is a local diffeomorphism onto the image G(S) near the point a ∈ S.

3. The hyperplane G(a) ⊂ Π transversely meets each curve πγi (∆i) at the point πγi (ti).

The next proposition provides a local, quantitative description of the first and the third conditions in the proposition above. We present it for the completeness since we will not employ it in this manuscript. Without loss of generality we assume that the intervals ∆ν contain the origin, and γν(0) = 0. Chapter 9. Hypersurfaces of double translation 116

Proposition 9.4. Let S ⊂ Rn+1 be a hypersurface having two representations as a trans-

lation hypersurface S = γ1 +...+γn and S = γn+1 +...+γ2n. Then there is a sufficiently small neighborhood of the origin in which S is a double translation hypersurface if and

only if

1. vectors γ˙ 1(0),..., γ˙ 2n(0) are pair-wise distinct,

2. none of the vectors γ¨1(0),..., γ¨2n(0) is tangent to S at the origin,

3. the second quadratic form of S at the origin is non-degenerate.

The next is a multidimensional variant of the example 3.

Example 4. Suppose that Γ ⊂ RP g−1 is the real part of a canonically embedded (see the next section for the terminology) non-hyperelliptic curve Γc ⊂ CP g−1 of genus g, and ` is a hyperplane meeting Γ transversely at 2g−2 points B1,...,B2g−2. Let Γ1,..., Γ2g−2 ⊂ Γ be the curves, with Bi ∈ Γi, swept out by all the hyperplanes in a connected neighborhood

g−1 ∗ U` of ` in the space of hyperplanes (RP ) . Suppose that

1. Curves Γν are disjoint and belong to the non-singular part of Γ.

2. Each Γi is a parameterized curve Γi : ∆i 7→ Γ defined on its own interval ∆i of the

real line. Each mapping Γi is injective and dΓi does not vanish on ∆i.

Assume that ω1, . . . , ωg is a real-valued basis, that is the restriction of each ωk to Γ is a real-valued 1-form, of the g-dimensional space of holomorphic 1-forms on Γc.

g−1 Let γ1, . . . , γg−1 ⊂ R be (g−1) curves parameterized by points of the curves Γ1,..., Γg−1 A A ! Rν Rν respectively in the following way: γν(Aν) = ω1,..., ωg , where Aν ∈ Γν. Bν Bν g−1 Let γg, . . . , γ2g−2 ⊂ R be (g−1) curves parameterized by points of the curves Γg,..., Γ2g−2 A A ! Rν Rν respectively in the following way: γν(Aν) = − ω1,..., ωg , where Aν ∈ Γν. Bν Bν Different choice of the basis ω1, . . . ωg leads to a linear transformation of the curves

g−1 γ1, . . . , γ2g−2 in R . Suppose that both γ1 + ... + γg−1 and γg + ... + γ2g−2 are smooth translation hypersurfaces in Rg. Chapter 9. Hypersurfaces of double translation 117

It follows from Abel’s theorem that γ1(A1) + ... + γg−1(Ag−1) = γg(Ag) + ... +

γ2g−2(A2g−2) for points A1,...,A2g−2 that belong to one hyperplane l ∈ U`. Indeed, due to connectivity of U` we may assume that l is sufficiently close to `. Indeed, Consider

1 T the projection π :Γ 7→ RP from the intersection plane ` l (it could be at infinity) to a line transversal to both hyperplanes ` and l. Let A and B be the images of all the points Ak and Bk respectively. According to Abel’s theorem, traceπω = 0 for ω a A A A2g−2 R R1 R generalized holomorphic form. In particular, traceπω = ω + ... + ω = 0. Thus B B1 B2g−2 γ1(A1) + ... + γg−1(Ag−1) = γg(Ag) + ... + γ2g−2(A2g−2).

g−1 Suppose that the linear span of any g − 1 points from B1,...,B2g−2 in RP is

RP g−2. It is then a direct corollary from Proposition 9.5 and Proposition 9.3 that in a sufficiently small neighborhood of the origin 1) S is a smooth surface having two representations S = γ1 + ... + γg−1 and S = γg + ... + γ2g−2 as a translation surface; 2) the two representations of S are “distinct”.

Duality between Lie’s construction and the example above.

Proposition 9.5 explains the interplay between Lie’s construction and the example 4. Let

ω1, . . . , ωN be smooth 1-forms on an interval ∆ ⊂ R that vanish simultaneously at no point of ∆. Having this data we can always construct two parameterized curves, one in

RP N−1 and another in RN .

First curve. Define the mapping F : ∆ 7→ RP N−1 by the following rule: F (t) is the point with homogeneous coordinates [f1(t): ... : fN (t)], where t is a coordinate on

∆ and ωk = fkdt. As the change of coordinates t = t(u) results in multiplication of

dt each function fk by the jacobian du , the mapping F is independent on the choice of a coordinate t.

Second curve. Now choose a point B ∈ ∆ and define a curve γ : ∆ 7→ RN by the formula  A A  R R γ(A) = ω1,..., ωN . B B N N−1 Now with the curve γ(∆) ⊂ R associate a parameterized curve πγ : ∆ 7→ RP by T N−1 the following rule: t ∈ ∆i is mapped into the intersectionγ ˙ (t) RP of the tangent Chapter 9. Hypersurfaces of double translation 118

lineγ ˙ (t) and the infinity hyperplane of the projective closure of RN .

Proposition 9.5. Curves πγ(∆) and F (∆) coincide.

t t ! R R Proof. Indeed, if t is a coordinate on ∆, then γ(A) = f1dt, . . . , fN dt . Thusγ ˙ (t) = t0 t0 N−1 (f1(t), . . . , fN (t)), and the mapping πγ : ∆ 7→ RP sends t to [f1(t): ... : fN (t)],

Below we state the complex-analytic analogue of Proposition 9.5. We will need it in

connection with Torelli theorem. Let ω1, . . . , ωN be holomorphic 1-forms on a connected

and simply-connected domain ∆ ⊂ C that vanish simultaneously at no point of ∆. As

N−1 in the real case we construct two parameterized curves πγ(∆) and F (∆) in CP and

CN , respectively.

Proposition 9.6. Curves πγ(∆) and F (∆) coincide.

Theorem 9.7 is Lie’s classification theorem of hypersurfaces of double translation and

Theorem 9.8 is its complex-analytic analogue.

Theorem 9.7. Let S = γ1 + ... + γn = γn+1 + ... + γ2n be a smooth hypersurface of

n+1 n double translation in R . Then 2n curves πγi (∆i) ⊂ RP in Lie’s construction belong to an algebraic curve Γ of degree 2n. Moreover, on the complexification Γc of the curve

Γ there are (at least) (n + 1) linearly independent over C generalized holomorphic forms.

Theorem 9.8. Let S = γ1 + ... + γn = γn+1 + ... + γ2n be an analytic hypersurface of

n+1 n double translation in C . Then 2n curves πγi (∆i) ⊂ CP in Lie’s construction belong to an algebraic curve Γ of degree 2n. Moreover, on Γ there are (at least) (n + 1) linearly independent over C generalized holomorphic forms.

Remark 9.2. In the analytic situation, under some extra genericity assumptions, Little

J [12] showed that, in fact, Γ is a canonically embedded connected (“non-hyperelliptic”)

Gorenshtein curve. Conversely, the degree of such a curve in CP g−1, where g is the dimension of the space of generalized holomorphic forms, is 2g − 2 and the construction in the example 4 leads to a hypersurface of double translation. Chapter 9. Hypersurfaces of double translation 119

Proof of the theorem 9.7. Let S = γ1(∆1)+...+γn(∆n) = γn+1(∆n+1)+...+γ2n(∆2n) be a hypersurface of double translation in Rn+1. Consider any linear functional l ∈ (Rn+1)∗.

On each curve πγk (∆k) ⊂ Π from Lie’s construction define a function by the following rule: fk(t) = l(γk(t)) if k = 1, . . . , n; fk(t) = −l(γk(t)) if k = n + 1,..., 2n. In other

−1 ∗ −1 ∗ words, fk = (πγν ) lk if k = 1, . . . , n and fk = −(πγν ) lk if k = n + 1,..., 2n.

Now if P = γ1(t1) + ... + γn(tn) = γn+1(tn+1) + ... + γ2n(t2n) then the hyperplane

G(P ) meets the curves πγk at 2n points, the sum of the values of fk at which is equal to zero.

If the curves γk = γk(∆k) were real-analytic, we would choose coordinate functions

n+1 ∗ l1, . . . , ln+1 ∈ (R ) so that each dlk does not vanish at one particular point on each curve. Then it follows from the analyticity, that each dlk vanishes on nowhere dense subsets on γ1, . . . , γ2n. The application of Theorem 7.1 to each lk would complete the proof. Since the curves are just smooth, we need some extra work.

n+1 ∗ Near a point P ∈ S choose coordinate functions l1, . . . , ln+1 ∈ (R ) so that, locally, dlk vanishes nowhere on γ1, . . . , γ2n. The application of Theorem 7.1 to each lk, asserts

n that near the line G(P ) ⊂ RP a) the curves πγν (∆ν) belong to an algebraic curve ΓP

c of degree 2n; b) there is a generalized holomorphic form on ΓP which restriction to each

−1 ∗ πγν (∆ν) coincide, up to the sign, with (πγν ) dlk. In particular, for each k the 1-forms

−1 ∗ (πγν ) dlk are real-analytic on πγν (∆ν), in the real-analytic structure of the curve ΓP . Since the above is arguments are valid for any point in S, we conclude that the curves

πγk (∆k) belong to an algebraic curve Γ of degree 2n and there are n + 1 generalized

c holomorphic 1-forms on Γ which restriction to each πγν (∆ν) coincide, up to the sign,

−1 ∗ with (πγν ) dlk for k = 1, . . . , n + 1.

As we showed the structure of a doubly translation hypersurface impose very strong

conditions on a hypersurface. The Proposition 9.9 below shows that, under an extra as-

sumption, there is no third translation structure on the given hypersurface. The Propo- Chapter 9. Hypersurfaces of double translation 120

sition 9.10 is the complex-analytic analogue. We say that S is a hypersurface of double

translation near a point a ∈ S if S shifted by the vector a is a hypersurface of double

translation.

Proposition 9.9. Under the notations of Theorem 9.7 assume that the algebraic curve

c Γ is irreducible. If S = α1 + ... + αn = αn+1 + ... + α2n is another representation of S

as a hypersurface of double translation, then γ1, . . . , γ2n is a permutation of α1, . . . , α2n.

Proposition 9.10. Under the notations of Theorem 9.8 assume that the algebraic curve

Γ is irreducible. If S = α1 + ... + αn = αn+1 + ... + α2n is another representation of S

as a hypersurface of double translation, then γ1, . . . , γ2n is a permutation of α1, . . . , α2n.

n Proof of Proposition 9.9. By Theorem 9.7 the disjoint union of 2n curves παi (∆i) in RP belongs to an algebraic curve Γ˜ of degree 2n. It is sufficient to show that Γ = Γ.˜ Assuming that these two curves are different, we find a point P ∈ S so that the hyperplane G(P ) transversely meets Γ S Γ˜ at 4n distinct points. By Proposition 9.3, near the point P the two translation structures of the hypersurface S = γ1 + ... + γn and S = α1 + ... + αn

n are “distinct”. Thus, by Theorem 9.7, the disjoint union of 2n curves παi , πγi ⊂ RP (i = 1, . . . , n) belongs to an algebraic curve Γ˜ of degree 2n. Since Γ is irreducible, it follows that Γ = Γ˜ = Γ.˜

9.3 Torelli theorem

Let S be a compact connected Riemann surface of genus g.

Canonical curve. It is the fundamental fact the vector space Ω1(S) of holomorphic

1 1-forms on S is g-dimensional (over C). With a basis ω1, . . . , ωg in Ω (S) one can asso- ciate the map F : S 7→ CP g−1 sending a point p ∈ S to the point with homogeneous coordinates [f1 : ... : fg], where ωk = fkdz, and z is a local coordinate near p ∈ S. As Chapter 9. Hypersurfaces of double translation 121

follows from Riemann-Roch theorem, forms ω1, . . . , ωg vanish simultaneously at no point of S and, for a non-hyperelliptic curve, the mapping F is injective. The local change of

coordinates z = z(w) results in the multiplication of all the functions fk by the jacobian

dz dw , so the mapping F does not depend on local coordinates involved in the construction.

We write F = [ω1 : ... : ωg]. Different choice of the basis ω1, . . . , ωg leads to a projective transformation in the image. The mapping F is called canonical embedding, and the

image of S is called a canonical curve. The curve F (S) ⊂ CP g−1 is non-degenerate (does not belong to a linear subspace of a lower dimension), algebraic of degree 2g − 2, and it is isomorphic to S.

Example 5. Let S ⊂ CP 2 be a smooth projective curve, and P be an irreducible polyno- mial of degree four that defines S in the affine chart with coordinates x, y. We have three

dx∧dy holomorphic forms ω, xω, yω, where ω = dP . Thus, F = [ω : xω : yω] = [1 : x : y] is the identity mapping.

Abel-Jacobi map. From the topological view-point S is a sphere with g handles.

1 Choose a basis γ1, . . . γ2g in the homology group H (S), and a basis ω1, . . . , ωg of holo- morphic 1-forms. Fix a point P0 on the curve S, and consider an oriented path γ from P0 to a point P in S. It is not unique, and is defined up to a cycle. Thus

R R g the vector ( γ ω1,..., γ ωg) ∈ C is defined up to a linear combination of 2g vectors (R ω ,..., R ω ) with integer coefficients. The resulting map Ab : S 7→ g\Λ is called γν 1 γν g C Abel-Jacobi map. In fact, the lattice generated by all the vectors Λ has a maximal rank, and thus the factor Cg\Λ is an g-dimensional compact complex torus J(S) (which is called a Jacobian for a symplectic choice of the basis in H1(S)). By linearity the mapping Ab extends to the mapping from any symmetric power S(k) to J(S) and, in particular, from

S(g−1). The image of S(g−1) under the Abel-Jacobi map is an analytic hypersurface θ(S)

(g−1) in J(S). A point P = {P1,...,Pg−1} ∈ S , where S is a non-hyperelliptic curve, is called admissible if the linear span < P > of the points F (P1),...,F (Pg−1) on the canoni- cal curve F (S) is (g−2)-dimensional. Lemma 9.13 implies that a generic, in Zariski sense, Chapter 9. Hypersurfaces of double translation 122

point in S(g−1) is admissible. Proposition 9.6 shows that the map Ab : S(g−1) 7→ J(S) has

the maximal rank at the admissible point P ; and the geometric form of the Riemann-

Roch theorem asserts that Ab(P ) has only one preimage – P . The next proposition shows

that θ(S) is a hypersurface of double translation near Ab(P ) ∈ θ(S) that coincide with

the complex-analytic version of the example 4 with the hyperplane ` taken as < P >.

Proposition 9.11. Let P ∈ S(g−1) be an admissible point and suppose that the other

g − 1 points of the intersection < P > T F (S) correspond to an admissible point in

S(g−1) as well. Then θ(S) is a hypersurface of double translation near the point Ab(P ).

Proof. We identify the curve S with its canonical image F (S). Let γ be an oriented

1-chain with the boundary ∂γ = P1 + ... + Pg−1 − nP0, where P0 is the fixed point

(g−1) and P = {P1,...,Pg−1. Take a point Q = {Q1,...,Qg−1} ∈ S close to P , and

consider an oriented 1-chain γQ that is the union of γ and small arcs γν connecting

points Pν with Qν (ν = 1, . . . , g − 1). The boundary ∂γQ = Q1 + ... + Qg−1 − nP0, and R R R R Ab(Q) = ( ω1,..., ωg). The latter differs by a constant vector ( ω1,..., ωg) γQ γQ γ γ R R R R (g−1) from the vector ( ω1 + ... + ω1,..., ωg + ωg). Now think of P,Q ∈ S γ1 γg−1 γ1 γg−1

as the hyperplanes determined by the linear span of points Pν and Qν, respectively, then, as Q varies near P , the later vector defines the hypersurface of double translation – the complex-analytic version of the example 4.

The image of S(g−1) under the mapping Ab is an analytic hypersurface in Cg\Λ. We shall say that a domain U ⊂ S(g−1) is sufficiently small if its image Ab(U) ⊂ Cg\Λ lifts up to an analytic hypersurface in a neighborhood of a point in Cg. The corresponding hypersurface is defined up to a translation in Cg and is denoted by Ab(U) as well. Now we are ready to state Torelli theorem

Theorem 9.12 (Torelli theorem). Two non-hyperelliptic compact Riemann surfaces S1 and S2 of genus g are isomorphic if and only if there are sufficiently small domains Chapter 9. Hypersurfaces of double translation 123

(g−1) (g−1) g g U1 ⊂ S1 and U2 ⊂ S2 , and an affine isomorphism I : C 7→ C that takes Ab(U1) to Ab(U2).

We will need the following well-known lemma

Lemma 9.13. Let Γ be a non-degenerate smooth algebraic curve in CP N . Then the linear span of any N − 1 points of a generic, in Zariski sense, hyperplane section is

CP N−1.

Proof of Theorem 9.12. The necessity is trivial. Now assume that there is an affine iso-

g g morphism I : C 7→ C that takes U1 ⊂ θ(S1) to U2 ⊂ θ(S2). Choose points P1 ∈ U1 and P2 ∈ U2 so that the corresponding hyperplanes < P1 > and < P2 > are generic in the sense of Lemma 9.13, and I(P1) = P2. Near the points P1 and P2 the hypersurfaces

θ(S1) and θ(S2) have double translation structure provided by Proposition 9.11. Since affine mappings preserve a translation structure of a hypersurface, by Theorem 9.10, the

2g − 2 curves in the first double translation hypersurface are mapped by I into 2g − 2 curves in the second double translation hypersurface.

Now the affine isomorphism I : Cg 7→ Cg induces a projective transformation I˜ in the infinity plane CP g−1. If S is a hypersurface of double translation in Cg, then Lie’s construction is commutative with I in the following sense: I˜(L(S)) = L(I(S)).

We conclude that the canonical curve of the first Riemann surface is mapped by I˜ into the canonical curve of the second Riemann surface. As we can clearly interchange ˜ S1 and S2, the mapping I is an isomorphism between the Riemann surfaces.

Remark 9.3. We will briefly indicate the connection of Theorem 9.12 with the Torelli

theorem in modern literature. The use generalities on linear bundles over compact com-

plex tori, Riemann’s theorem, and the fact that any biholomorphic isomorphism between

complex (compact) tori is linear reduces Torelli theorem as, say, in [7] to the following

statement. Two non-hyperelliptic compact Riemann surfaces S1 and S2 of genus g are Chapter 9. Hypersurfaces of double translation 124

isomorphic if and only if there is linear isomorphism L : J(S1) 7→ J(S2) between the ja-

cobians of the curves that takes the analytic hypersurface θ(S1) ⊂ J(S1) to the analytic hypersurface θ(S2) ⊂ J(S2) up to a shift in J(S2).

Remark 9.4. Theorem 9.12 naturally generalizes to the case of two non-hyperelliptic irreducible Gorenshtein curves. The proof remains unchanged. Chapter 10

Appendix

The aim of this appendix is to prove Proposition 10.1 below which we partially need in

the proof of sufficiency in our second main theorem.

Theorem 10.1. Let Γ be an n-dimensional algebraic variety in an affine space and ω

an n-form on Γ holomorphic outside of a hypersurface that contains the singular locus of

Γ. Then the following three conditions on the form ω are equivalent:

1. There is a proper finite polynomial mapping f :Γ 7→ Cn so that for any polynomial

g with complex coefficients the trace of gω under the mapping f is Pgdy1 ∧ ... ∧ dyn

n with Pg ∈ C[y1, . . . , yn] a polynomial in a fixed affine coordinates y1, . . . , yn in C .

2. The trace of ω under any proper finite polynomial mapping F :Γ 7→ Cn is of

the form PF dy1 ∧ ... ∧ dyn with PF ∈ C[y1, . . . , yn] a polynomial in a fixed affine

n coordinates y1, . . . , yn in C .

3. For any polynomial g with complex coefficients the trace of ω under any proper

n finite polynomial mapping F :Γ 7→ C is PF dy1 ∧ ... ∧ dyn with PF ∈ C[y1, . . . , yn]

n a polynomial in a fixed affine coordinates y1, . . . , yn in C .

The idea is, first, to prove the theorem for subvarieties of complete intersections, see

Lemma 10.2 below, and then to embed an arbitrary affine variety into a complete inter-

125 Chapter 10. Appendix 126 section, see Lemma 10.3 below. In Lemmas 10.2, 10.3, a generic point is a generic point in the sense of Zariski topology.

N Lemma 10.2. Let D1,...,DN−n be algebraic hypersurfaces in C meeting each other T T transversely at a generic point of an n-dimensional algebraic variety Γ ⊆ D1 ... DN−n. Suppose that ω is an n-form on Γ that satisfies the first condition of Theorem 10.1. Then

Qdx1∧...∧dxN there is a polynomial Q so that ω = , where Pν ∈ [x1, . . . , xN ] is the poly- dP1∧...∧dPN−n C nomial without multiple irreducible factors that defines Dν.

Lemma 10.3. Let F :Γ 7→ Cn be a proper finite polynomial mapping between an n- dimensional affine algebraic variety and Cn. Then, for some N, there is an n-dimensional algebraic variety Γ˜ ⊂ CN such that:

1. The variety Γ˜ ⊂ CN is a complete intersection, that is there are algebraic hyper-

N surfaces D1,...,DN−n in C meeting each other transversely at a generic point of ˜ T T Γ = D1 ... DN−n.

2. There is polynomial mapping π : Γ˜ 7→ Γ that sends a subvariety of Γ˜ isomorphically,

in the sense of algebraic varieties, to Γ.

3. There is a proper finite polynomial mapping F˜ : Γ˜ ⊂ Cn such that the following

i diagram commutes Γ / ˜ @ Γ @@ ~~ @@ ~~ F @@ ~~F˜ @ ~~ Cn where i :Γ 7→ Γ˜ is an inclusion that is the inverse of π restricted to a certain

subvariety of Γ˜.

Proof of Theorem 10.1. Since the implications 3 ⇒ 2 and 3 ⇒ 1 are trivial, it is sufficient to show that 1 ⇒ 2, 3 and 2 ⇒ 1.

We start with the implications 1 ⇒ 2, 3. Let ω be an n-form on Γ that satisfies

Theorem 10.1(1) with f :Γ 7→ Cn a fixed proper finite polynomial mapping, and let Chapter 10. Appendix 127

F :Γ 7→ Ck be an arbitrary proper finite polynomial mapping. Define the n-formω ˜ on the complete intersection Γ,˜ provided by Lemma 10.3, as follows: on the component

i(Γ) isomorphic to Γ it is (i−1)∗ω and it is zero otherwise. The property of a mapping between two varieties to be proper is intrinsic and, in particular is independent on their embeddings, so f ◦ π : i(Γ) 7→ Cn is proper finite polynomial mapping. The notion of trace is intrinsic as well. We conclude that tracef◦πω˜ = tracef ω.

Qdx1∧...∧dxN By Lemma 10.2,ω ˜ = , where P1,...,PN−n ∈ [x1, . . . , xN ] are irreducible dP1∧...∧dPN−n C ˜ T T polynomials defining the hypersurfaces D1,...,DN−n with Γ = D1 ... DN−n and Q

a polynomial in variables x1, . . . , xN with complex coefficients.

Let g be any polynomial with complex coefficients. Since the formω ˜ is identically

equal to zero outside i(Γ), the application of Corollary 3.31 to the complete intersection

Γ,˜ form (i−1)∗gω, and the proper finite polynomial mapping F : Γ˜ 7→ Cn completes the proof of the implications 1 ⇒ 2, 3.

We now show the implication 2 ⇒ 1. Suppose that ω is the highest degree form

on an n-dimensional affine variety Γ ⊂ CN and the trace of ω under any proper finite

N polynomial mapping is polynomial. Choose an (N − n)-dimensional plane L0 ⊂ C that transversely meet Γ in d points with d the degree of the variety, and an n-dimensional

N N plane M ⊂ C so that C is isomorphic to L0 × M. Consider the projection π of

L0 × M onto the second multiplier. As follows from the implicit function theorem, there

T −1 T is a small connected neighborhood U ⊂ L0 of the point L0 Γ so that π (U) Γ is a disjoint union of surfaces as in our main theorems.

Now a parallel projection πL along an (N −n)-dimensional plane L that is sufficiently

close to L0 is a proper finite polynomial mapping, and thus traceπL = PLdz1 ∧ ... ∧ dzn,

where z1, . . . , zn are affine coordinates in L0 and PL ∈ C[z1, . . . , zn] is a polynomial of

degree bounded above by a constant independent of πL. The latter is due to necessity of Theorem 5.4. Proposition ?? ensures that, locally, we can apply Proposition 6.8. Thus,

in particular, for any polynomial g ∈ C[x1, . . . , xN ] the trace of gω under the mapping Chapter 10. Appendix 128

π|Γ is polynomial.

Proof of Lemma 10.3. Consider the graph Γ˜ ⊂ CN × Cn of F . The projection of CN ×

Cn onto the second multiplier is a covering over a Zariski open subset of Cn and the application of 2.16 completes the proof.

For the proof of Lemma 10.2 we need the following proposition. Let K be either C or

n R. Suppose that A1,...,Ak ∈ K are non-degenerate solutions (not necessarily all) of

n a polynomial system of equations P1 = ... = Pn = 0 with Pk ∈ K [x1, . . . , xn]. I recall

n that a solution A ∈ K is non-degenerate if the jacobian JP of the functions P1,...,Pn is not equal to zero at A.

Proposition 10.4. There is a polynomial H ∈ K[ξ, x] in 2n variables ξ = (ξ1, . . . , ξn) j j and x = (x1, . . . , xn) such that H(Ai,Aj) = JP (Ai)δi , where δi is the Kronecker symbol. Moreover, the polynomial H remains unchanged for non-generate solutions of the system

P1 = a1,...,Pn = aa with arbitrary ak ∈ K.

Proof of Lemma 10.2. For a proper finite polynomial mapping f :Γ 7→ Cn we choose a n −1 S connected domain U ⊂ C so that f (U) = Uν is d-sheeted over U. We denote by

−1 A1(p),...Ad(p) the preimages f (p) of a point p ∈ U.

The application of Lemma 10.4 to P1,...,PN−n, f1, . . . , fn yields to a polynomial H =

P α α α1 αn j cαx with cα ∈ K[ξ] and x = x1 . . . xn such that: H(Ai(p),Aj(p)) = JP (Ai(p))δi , where JP is the Jacobian of the functions P1,...,PN−n, f1, . . . , fn with respect to the variables x1, . . . , xN .

Now, let ρk be a function defined from the identity ω|Uk = ρkdf1 ∧ ... ∧ dfn. Consider the function Q on U × CN given by the formula

Q(y, x) = ρ1(A1(y))H(A1(y), x) + ... + ρd(Ad(y))H(Ad(y), x). Chapter 10. Appendix 129

j If a is a point in some Uk, then the property H(Ai(p),Aj(p)) = JP (Ai(p))δi applied to

−1 f (f(a)) = {a, a1, . . . , ad−1} yields to

Q(f(a), a) = ρk(a)JP (a),

where JP is the Jacobian of the functions P1,...,PN−n, f1, . . . , fn with respect to the

variables x1, . . . , xN .

Note that, in fact, Q coincide with a polynomial on CN+n. Indeed,

X α Q(y, x) = [ρ1(A1(y))cα(A1(y)) + ... + ρd(Ad(y))cα(Ad(y))]x ,

α so the coefficient before x is the polynomial Pcα in the coordinate expression of the trace of cα(x)ω under the mapping f. We conclude that Q is a polynomial. I claim that Q(f, x) is the polynomial we are looking for. From the identity Q(f(a), a) =

Q(f,x)dx1∧...∧dxN ρk(a)JP (a) above, it directly follows that |U = ρν. We conclude dP1∧...∧dPN−n∧df1∧...∧dfn ν

Q(f,x)dx1∧...∧dxN that |U = ρνdf1 ∧ ... ∧ dfn = ω|U dP1∧...∧dPN−n ν ν

Remark 10.1. It directly follows from the formula for the polynomial H ∈ C[ξ, x] given

in the proof of Proposition 10.4 that degH ≤ degP1 + ... + degPn − n. Now denoting the

polynomials Pg in Theorem 10.1 by [tracef gω] we rewrite the formula for the polynomial Q in the following beautiful way:

X ∗ α Q = f [tracef cαω]x , |α|≤degH

where |α| = α1 + ... + αn and degcα ≤ degP1 + ... + degPn − n − |α|.

We now proceed with the proof of the Proposition 10.4

n Proof of Proposition 10.4. For any smooth functions P1,...,Pn : C 7→ C the application of Adamar’s lemma ensures existence of an n × n function-valued matrix P with entries Chapter 10. Appendix 130

n n Pij : C × C 7→ C smooth functions so that       P1(x) − P1(ξ) P11 P12 P13 ...P1n x1 − ξ1             P2(x) − P2(ξ) P21 P22 P23 ...P2n x2 − ξ2    =     , (10.1)  .   . . . . .   .   .   ......   .              Pn(x) − Pn(ξ) Pn1 Pn2 Pn3 ...Pnn xn − ξn

n n where (x, ξ) = (x1, . . . , xn, ξ1, . . . , ξn) are coordinates in C × C . In addition, the fol-

∂Pi lowing identity holds Pij(x, x) = (x). ∂xj

If, however, P1,...,Pn are polynomials then the entries Pij are polynomials as well.

Indeed, for any smooth function f : Cn 7→ C and x, ξ ∈ Cn the following identity is 1 R d valid: f(x) − f(ξ) = dt f(ξ + t(x − ξ))dt. Using the chain rule and interchanging the 0 integration and differentiation, we obtain

1 1 Z ∂ Z ∂ f(x) − f(ξ) = (x1 − ξ1) f(ξ + t(x − ξ))dt + ... + (xn − ξn) f(ξ + t(x − ξ))dt. ∂x1 ∂xn 0 0

Note that if P1,...,Pn are polynomials with real coefficients, then the polynomials Pij also have real coefficients.

I claim that the polynomial H = det P satisfy all the properties in Proposition 10.4.

n Let A1,...,Ak ∈ C be non-degenerate solutions of a polynomial system of equations

P1 = ... = Pn = 0. Substitute x = A1 and ξ = Aj with j 6= 1. It follows that the matrix

P (A1,Aj) has a non-zero eigenvector with coordinates A1 − Aj. Thus det P (A1,Aj) = 0.

By the property of polynomials Pij, the matrix P evaluated at (A1,A1) is the Jacobian matrix of the functions P1,...,Pn at A1. Finally, note that the matrix P is independent on the choice of a system P1 = a1,...,Pn = aa.

Remark 10.2. One can consider a more general situation. Suppose that coefficients of

P1,...,Pn ∈ C[x1, . . . , xn] polynomially depend on a set parameters. Then there exists a polynomial H in 2n variables with coefficients polynomially depending on the same set of parameters such that it has the same properties as H from Proposition 10.4. Chapter 10. Appendix 131

Together with the technique in the proof of Lemma 10.2, the use of the polynomial H that polynomially depends on parameters leads to further generalizations of Theorem 10.1.

We state one of them. In fact, Theorem 10.5 implies the theorem stated in the very end of Chapter 3.1. By holomorphic mapping below we mean a holomorphic mapping in the sense of algebraic varieties.

Theorem 10.5. Let Γ be an n-dimensional algebraic variety and ω an n-form holomor- phic on the complement to a hypersurface that contains the singular locus of Γ. Suppose that f : U 7→ V is a proper finite holomorphic mapping between Zariski open subsets

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