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.