GEOMETRY OF SYMMETRIC POWERS OF COMPLEX DOMAINS

Christopher Grow

A thesis submitted in partial fulfillment of the requirements for the degree of Master of Arts

Department of Mathematics

Central Michigan University Mount Pleasant, Michigan April 2017 ABSTRACT

GEOMETRY OF SYMMETRIC POWERS OF COMPLEX DOMAINS

by Christopher Grow

m Given a complex manifold X, its m-fold symmetric power XSym is the collection of all unordered m-tuples of elements of the complex manifold. Such symmetric products arise naturally in complex analysis, for example as the set of solutions of a polynomial equation. The symmetric powers inherit a complex structure, so that it is possible to define holomorphic functions and maps on them. In this work, our goal is to understand aspects

m of the complex geometry of the symmetric powers (Bs)Sym, where Bs is the open unit ball in

s m the complex vector space C , by classifying the proper holomorphic maps from (Bs)Sym to itself. These proper holomorphic self-maps are distinguished by the fact that they carry the boundary of a space to itself, and are a natural generalization of the notion of biholomorphic automorphisms.

In the case when s = 1, the m-th symmetric power of the unit disc in C is known traditionally as the m-dimensional symmetrized polydisc, and has been intensively studied in view of its importance in operator theory and control engineering. The symmetrized polydisc

can be thought of as an open subset of Cm. Edigarian and Zwonek (2005) determined the proper holomorphic self-maps of the symmetrized polydisc, and showed that these self-maps arise from proper holomorphic self-maps of the unit disc. The main result of this thesis is a generalization of the result of Edigarian and Zwonek

m to the symmetric power (Bs)Sym of the unit ball Bs, where s ≥ 2. As a first step, we show

m N that (Bs)Sym can be identified with an open subset of an affine algebraic variety in C for a large integer N depending on m and s. This is done by using a symmetric analog of the classical Segre Embedding of a product of projective varieties in a higher dimensional

m projective space. This gives an elementary way to endow (Bs)Sym with a complex structure.

m Then we reduce the problem of classifying proper holomorphic self-maps of (Bs)Sym to a ii certain lifting problem, the obstruction to solving which lies in the fundamental group of

m m a certain open subset of (Bs) , obtained by removing an analytic set from (Bs) . Using a transversality argument, we show that this obstruction in fact vanishes, thus proving the

m result. As a consequence, we find that each proper holomorphic self-map of (Bs)Sym is an automorphism induced by an automorphism of Bs.

iii TABLE OF CONTENTS

CHAPTER

I. INTRODUCTION ...... 1 I.1. The Symmetrized Polydisc ...... 1 I.2. Holomorphic Maps and the Theorem of Edigarian and Zwonek ...... 2 I.3. Proper Holomorphic Mappings of Higher Dimensional Symmetric Powers .4 I.4. Topological Properties of Proper Maps ...... 8 I.5. Some Examples of Proper Holomorphic Maps ...... 10

II. ANALYTIC SETS AND PROPER MAPS ...... 13 II.1. Analytic Subsets of Complex Manifolds ...... 13 II.2. Proper Holomorphic Maps ...... 15

III. SYMMETRIC POWERS ...... 21 III.1. Definitions and Preliminaries ...... 21 III.2. Topological Symmetric Powers ...... 22 III.3. Algebraic Embedding of Symmetric Powers of Projective Spaces ...... 25 III.4. Embedding of Symmetric Powers of Affine Spaces and of Balls ...... 31 III.5. Complex Symmetric Powers ...... 34 III.6. Symmetric Powers as Complex Spaces ...... 37

IV. PROPER HOLOMORPHIC SELF-MAPS OF SYMMETRIC POWERS OF BALLS 39 IV.1. Parametrized Analytic Sets ...... 39 IV.2. Some Results of Remmert-Stein Theory ...... 40 IV.3. Results of Edigarian and Zwonek ...... 44 IV.4. Mappings from Cartesian to Symmetric Powers ...... 47 IV.5. End of the Proof of Theorem 3 ...... 50

REFERENCES ...... 52

iv CHAPTER I INTRODUCTION

I.1. The Symmetrized Polydisc

Let D ⊂ C be the unit disc: D = {z ∈ C| |z| < 1}. The subset Σ2D of C2 given by

2 2 Σ D = {(z + w, zw) ∈ C |z, w ∈ D} is called the Symmetrized Bidisc. It can be shown that this is a domain, by which we mean an open connected set, which is pseudoconvex but has nonsmooth boundary. This domain first emerged in the study of certain problems in Electrical Engineering (see [1]). However, the symmetrized bidisc is a natural object to consider in complex analysis of several variables. Consider, for example, a quadratic equation in t with complex coefficients s and p:

t2 − st + p = 0 such that both roots z, ω of this equation lie in the unit disc. Then clearly the coefficients

(s, p) lie in Σ2D. It is possible to generalize this construction, and define the m-dimensional Sym-

m th metrized polydisc Σ D in the following way. For 1 ≤ k ≤ m, denote by σk the k elementary symmetric polynomial in m variables. These polynomials are defined to be σ1(z1, . . . zm) = Pm P j=1 zj, σ2(z1, . . . , zm) = j

m m Σ D = {(σ1(z), σ2(z), . . . , σm(z)) ∈ C |z = (z1, . . . , zm), where each zj ∈ D}.

m m As before, a point (s1, . . . , sm) ∈ C lies in Σ D if and only if all the roots of the polynomial equation

m m−1 m−2 m t − s1t + s2t − · · · + (−1) sm = 0

1 lie in D. Since the appearance of [1], the domains ΣmD have been a subject of much study from the complex analytic point of view (see e.g. [2, 3, 4, 11, 12, 20, 14, 15, 8]). Perhaps the most remarkable property of these domains is that though they are proper holomorphic images of convex domains, the theorem of Lempert on the equality of the Kobayashi and Carathedory metrics on convex domains does not extend to them (see [11]).

I.2. Holomorphic Maps and the Theorem of Edigarian and Zwonek One of the avenues of this work has been to explore mapping properties and symme-

tries of the domain ΣmD. Such questions regarding holomorphic mappings are very natural and important in complex analysis. Suppose U and V are open subsets of Cn. If we have a bijective holomorphic map f : U → V then f −1 is also holomorphic. In this case, we call f a biholomorphism. One of the famous theorems of mathematics is the Riemann Mapping The-

orem which states that each simply connected proper subdomain of C is biholomorphic to the unit disc D. It is also well-known how to determine the group Aut(D) of all automorphisms of D (i.e. biholomorphisms from D to itself). For domains in Cn with n ≥ 2, the study of holomorphic maps is much more compli- cated, and there is no higher-dimensional analog of the general theory of conformal mapping. In particular, there is no simple analog of the Riemann Mapping Theorem: not all open,

simply-connected proper subsets of Cn are biholomorphic to the unit ball

n 2 2 1 Bn := {z ∈ C : |z| = (|z1| + ··· + |zn| ) 2 < 1} (I.1)

2 when n ≥ 2. The simplest counter-example is the domain D = D × D, where D = B1, the

2 unit disc. Although D is in fact homeomorphic to B2, there is no biholomorphism between these domains, a fact known to Poincar´e. We give a proof of this result based on a more general theorem of Remmert and Stein (see [21]) in Chapter IV. In spite of the difficulties in studying general biholomorphic mappings of several

2 variables outlined above, it is still sometimes possible to characterize mappings between special domains. In this thesis, which is motivated by the results of Jarnicki, Pflug, Edigarian, and Zwonek, which we describe below, we study the structure of interesting holomorphic maps between some higher dimensional analogs of the symmetrized polydisc. We begin by summarizing the known results about holomorphic maps between symmetrized polydiscs, on which our work is based. The first step in this direction is [20], in which Jarncki and Pflug explicitly determined the group of automorphisms of the symmetrized polydisc. We do not state this result sepa- rately, since it is a special case of the results of Edigarian and Zwonek on proper holomorphic maps of these domains, which we will now describe. A proper map is a weakening of the notion of a homeomorphism between topological spaces. For f to be a proper map, instead of requiring that f is bijective and has a contin- uous inverse, we require that the inverse image of a compact set is compact. Clearly, each homeomorphism is a proper map, but not every proper map is a homeomorphism. Here, as well as in subsequent chapters, by a map, we mean a continuous function between topological spaces. In complex analysis of one variable, proper holomorphic maps arise naturally as “branched coverings”. A typical example of a proper holomorphic map is given by the map f from the unit disc D to itself given by

f(z) = zd where d is a positive integer. Notice that (1) the map is proper, since the inverse image of any closed subdisc {|z| ≤ r} of D (where 0 < r < 1) is a closed subdisc of D, (2) the map f takes the boundary of D to itself, and (3) on D \{0}, f is a covering map of d sheets. These are illustrations of general properties of proper holomorphic maps. See Section I.4 below for more examples and properties of proper holomorphic maps.

3 With these preliminaries, we can state the following result, to the generalization of which this thesis is devoted:

Theorem 1. ([14, 15]) Let f :ΣmD → ΣmD be a proper holomorphic map, and let σ =

m m (σ1, . . . , σm): C → C . Then, there exists a proper holomorphic map B : D → D such that

f(σ(z1, . . . , zm) = σ(B(z1),...,B(zm)).

Remark. One can easily characterize all proper holomorphic maps D → D. See Example 1.

This theorem demonstrates that each proper holomorphic self-map of ΣmD is induced by a proper holomorphic self-map of D. Also note that, as a special case, this shows that every automorphism of ΣmD is induced by an automorphism of D, thus recapturing the main result of [20].

I.3. Proper Holomorphic Mappings of Higher Dimensional Symmetric Powers In order to state the generalization of Theorem 1, we will first need to understand the domains ΣmD intrinsically. To do this, given a set Z, we form what is called its m-

m fold symmetric power, denoted by ZSym. This is the collection of all unordered m-tuples of

3 elements of Z. A typical element of ZSym is of the form

hz1, z2, z3i where z1, z2, z3 ∈ Z and the ordering of the elements is irrelevant, i.e.,

hz1, z2, z3i = hz2, z1, z3i = hz3, z1, z2i = hz3, z2, z1i = ...

Here and throughout this thesis, we adopt the convention of denoting elements of tuples taken from the same set by superscripts. In a very few cases when exponents are necessary, the meaning will be clear from the context.

m The symmetric power ZSym should be contrasted with the better known Cartesian

4 power Zm of ordered m-tuples. Recall that this means for example that in Z3, the two elements (z1, z2, z3) and (w1, w2, w3) are equal if and only if z1 = w1, z2 = w2 and z3 = w3. Note also that if we are given a mapping of sets

f : Z → W it induces in a natural way a mapping

m m m fSym : ZSym → WSym given by

m 1 2 m 1 m m fSym(hz , z , . . . , z i) = hf(z ), . . . , f(z )i ∈ WSym.

m This is easily seen to be well-defined. We call fSym the m-th symmetric power of the map f. In order to see the relevance of this notion, consider now the map Φ from the symmet-

m m 1 m ric power DSym of the unit disc to the symmetrized polydisc Σ D given, for z , . . . , z ∈ D by

1 m 1 m 1 m 1 m
m Φ(hz , . . . , z i) = σ1(z , . . . , z ), σ2(z , . . . , z ), . . . , σm(z , . . . , z ) ∈ C , where σk denotes the k-th elementary symmetric polynomial in m variables.

m m Proposition 1. Φ is a well defined bijection from DSym onto Σ D.

Proof. Since each component of Φ is a symmetric polynomial in z1, . . . , zm, Φ is invari- ant under permutation of z1, . . . , zm, hence is well-defined. Furthermore, given an element

m 1 m 1 m 1 m w ∈ Σ D, we have w = (σ1(z , . . . , z ), . . . , σm(z , . . . , z )) for some z , . . . , z ∈ D. The unordered tuple hz1, . . . , zmi are precisely the roots of the polynomial

m 1 m m−1 m 1 m p(t) = t − σ1(z , . . . , z )t + ··· + (−1) σm(z , . . . , z ), which are uniquely defined, up to permutation. This shows that Φ is a bijection.

In view of the above proposition, we can identify the symmetrized polydisc with the

5 m set DSym. With this understanding, Theorem 1 can be stated in the following way using the terminology introduced above.

m m Theorem 2. Let f : DSym → DSym be a proper holomorphic map. Then there is a proper

m holomorphic map g : D → D such that f = gSym.

We can now explain the main new results of this thesis. Consider the m-fold sym-

m s metric power (Bs)Sym of the unit ball Bs ⊂ C (see (I.1)). In the case that s = 1, we have

m B1 = D, the unit disc, and we already saw in Proposition 1 that DSym can be identified naturally with the open subset ΣmD of Cm. This allows us to talk about holomorphic maps

m from DSym to itself. In Chapter III we will show that a similar statement is true for the symmetric power

m (Bs)Sym when s ≥ 2. We construct an injective continuous map (which we call the Segre-

m N Whitney embedding) which embeds (Bs)Sym into a complex vector space C for some large N as a local analytic subset. Consequently, it makes sense to speak about holomorphic maps

m m from (Bs)Sym into (Bs)Sym. We note here that the explicit construction of the Segre-Whitney map in Chapter III can be avoided if we were to use the theory of complex analytic spaces that, informally, can be thought of as complex manifolds with singularities, modeled after analytic sets. Using

m general results in this theory, one can then construct (Bs)Sym directly as a quotient of the

m sm domain (Bs) in C under the action of the symmetric group Sm. This is explained briefly

m in Section III.6. We prefer to use the simpler description of (Bs)Sym as a certain subset of a Euclidean space.

m m Theorem 3. Let s ≥ 2, m ≥ 2, and let f :(Bs)Sym → (Bs)Sym be a proper holomorphic

map. Then, there exists a holomorphic automorphism g : Bs → Bs such that

m f = gSym,

that is f is the m-th symmetric power of g. 6 j Recall that for z ∈ Bs, j = 1, . . . , m we have

m 1 m 1 m gSym(hz , . . . , z i) = hg(z ), . . . , g(z )i.

Note the difference between Theorem 2 and Theorem 3: When s ≥ 2, each proper

m holomorphic self-mapping of (Bs)Sym is induced by an automorphism of Bs, which gives us the following Corollary. This is because, as we explain in Section I.5, each proper holomorphic self-mapping of Bs is an automorphism when s ≥ 2.

m Corollary 1. Each proper holomorphic self-map of (Bs)Sym is an automorphism, and is of

m the form gSym, where g is an automorphism of the ball Bs.

Since it is possible to describe all automorphisms of the ball explicitly (see (I.2) below), this gives a complete and explicit answer to the question of determining all proper

m holomorphic self-maps of (Bs)Sym. In Chapter IV, we prove Theorem 3 by first classifying all proper holomorphic maps

m m (Bs) → (Bs) using a Remmert-Stein type argument, which is the content of Theorem 9.

m m m m Then, we show that we can lift a map f :(Bs) → (Bs)Sym to a map fe :(Bs) → (Bs) ,

m by considering the fundamental group of (Bs) after removing certain analytic sets. Then, using Theorem 9, we can immediately deduce the structure of fe and hence, f. We then use

m m our classification of proper holomorphic maps (Bs) → (Bs)Sym to deduce the structure of

m m all proper holomorphic maps (Bs)Sym → (Bs)Sym. Theorem 3 can be immediately generalized to the situation in which the unit ball

s Bs is replaced by a strongly pseudoconvex domain in C . With a little more work, we can also generalize it to situations where Bs is replaced by a smoothly bounded pseudoconvex domain. For simplicity, we prefer to state the result in the form above.

7 I.4. Topological Properties of Proper Maps In this section, we collect some properties of proper maps between topological spaces which will prove useful later.

Definition 1. A map from a topological space X to a topological space Y is called proper if the preimage of each compact set in Y is compact in X.

We note here a few elementary properties of proper maps.

Proposition 2. Let X and Y be Hausdorff spaces and Z a subspace of Y . Let f : X → Z

∞ ∞ be a proper map. If a sequence {xn}n=1 ⊂ X has no limit point in X, then {f(xn)}n=1

∞ has no limit point in Z. Additionally, if Z is compact, then {f(xn)}n=1 has a subsequence converging to a point in ∂Z.

∞ Proof. Let f : X → Z a proper map and let {xn}n=1 ⊂ X be a sequence with no limit

∞ point in X. Consider the sequence {f(xn)}n=1. Suppose this sequence has some limit

∞ point z ∈ Z. Then, there is a subsequence {f(xnk )}k=1 converging to z, and the set

∞ −1 ∞ {f(xnk )}k=1 ∪ {z} is compact in Z. Hence, since f is proper, f ({f(xnk )}k=1 ∪ {z}) is

∞ −1 ∞ ∞ compact in X. Since {xnk }k=1 ⊂ f ({f(xnk )}k=1 ∪ {z}), this means {xnk }k=1 has a limit

−1 ∞ point in f ({f(xnk )}k=1 ∪ {z}), and hence in X. Consequently, {xn} has a limit point in

∞ X. Since no such limit point exists in X, this shows that {f(xn)}n=1 has no limit point in

∞ Z. If Z is compact, then {f(xn)}n=1 must have a limit point in Z. Since this limit point cannot be in Z, it must be in ∂Z.

Proposition 3. Let X and Y be locally compact Hausdorff spaces and f : X → Y a proper map. Then f is a closed map.

Proof. Let E ⊂ X be closed. Let y ∈ Y \ f(E). Since Y is locally compact, there is an open neighborhood U of y in Y with compact closure. Since f is proper, f −1(U) is compact. Since E is closed, K = f −1(U) ∩ E is compact. Thus, f(K) ⊂ f(E) is compact, and hence

8 closed in Y . Let V = U \ f(K). Then V is an open neighborhood of y, disjoint from f(E). Hence, f(E) is closed.

Proposition 4. Let X and Y be topological spaces and f : X → Y a proper map. Let V be a subspace of Y , set U = f −1(V ) and let g be the restriction of f to U. Then g : U → V is a proper map.

Proof. Let K ⊂ V be a compact set, so that K is also a compact subset of Y . By construction g−1(K) = f −1(K), so g−1(K) is a compact subset of U.

Proposition 5. Let X, Y , and Z be topological spaces and maps g : X → Y , f : Y → Z such that f ◦ g is a proper map. Then g is a proper map and the restriction of f to g(X),

f|g(X) : g(X) → Z, is a proper map.

Proof. Let K ⊂ Y be compact. Then, since f is continuous, f(K) is compact, and hence, (f ◦ g)−1(f(K)) = g−1((f −1 ◦ f)(K)) is compact as well. Since K ⊂ (f −1 ◦ f)(K), g−1(K) ⊂ g−1((f −1 ◦ f)(K)). Thus, since g is continuous, g−1(K) is a closed subset of a compact set, and therefore, compact. Hence, g is proper. Now, let K ⊂ Z be compact. Then (f ◦g)−1(K) is compact, and since g is continuous,

−1 −1 −1 so is g((f ◦ g) (K)) = f (K) ∩ g(X) = (f|g(X)) (K). Hence, f|g(X) is proper.

Proposition 6. Let X and Y be locally compact Hausdorff spaces and f : X → Y a proper bijective map. Then, f is a homeomorphism.

Proof. By Proposition 3, f is a closed map. Since f is a bijection, for any open set U ⊂ X, we have f(X \ U) = f(X) \ f(U) = Y \ f(U) is closed in Y . Hence, f(U) is open in Y . Thus, f is an open map, and so f −1 is continuous.

9 I.5. Some Examples of Proper Holomorphic Maps Like a proper map is a generalization of a homeomorphism, a proper holomorphic map f : M → N between complex manifolds can be thought of as a generalization of a biholomorphism, that is, a holomorphic map with a holomorphic inverse. Since every biholomorphism is a homeomorphism, a biholomorphism is necessarily proper. However, as before, it is not always the case that a proper holomorphic map is a biholomorphism. In the following examples, we compute the proper holomorphic self-mappings of certain domains

in Cn. Let D denote the open unit disc: {z ∈ C : |z| < 1}.

Example 1. Let f : D → D be a proper holomorphic map. Since f is proper, f must be

−1 −1 n nonconstant, and f (0) compact. Hence, f (0) is finite. Let {αi}i=1 be the zeros of f, allowing for repetition when a zero has multiplicity. Let

n Y z − αi g(z) = 1 − α z i=1 i for all z ∈ D. Note that for |z| = 1, we have

z − αi z − αi = = 1. 1 − αiz z(z − αi)

Hence, by the maximum modulus principle, |g(z)| ≤ 1 for all z ∈ D. Now, since f and g f have exactly the same zeros with the same multiplicities, g has only removable singularities in D, and therefore can be extended to a holomorphic function on D with no zeros in D.

∞ Since f is proper, by Proposition 2, if {ζn}n=1 is any sequence in D converging to a point in ∂D, we must have

lim |f(ζn)| = 1. n→∞

Since the same is true for g, we have

f(ζn) lim = 1. n→∞ g(ζn)

10

Hence, by the maximum modulus principle, we have f ≤ 1 on . Now, since f has no g D g

zeros on , f cannot have a local minimum in either. Hence, f ≡ 1 on . Hence, we D g D g D have proved that if f : D → D is a proper holomorphic map, then

n Y z − αi f(z) = eiλ 1 − α z i=1 i for some λ ∈ R and α1, . . . , αn ∈ D.

Example 2. Let f : C → C be a proper holomorphic map. Since f is entire, there is a power series expansion of f at the origin

∞ X n f(z) = anz , n=0 which converges on the whole plane. Now, consider the function

∞ n 1 X 1 F (z) = f = a . z n z n=0 This gives a Laurent series expansion of F (z) at the origin. Since the power series for f converges on the whole plane, the Laurent series for F converges on the punctured plane

C \{0}. Suppose the series for F does not terminate. Then, F has an essential singularity at 0, which means that f has an essential singularity at ∞. Hence, f(C \ Dr) is dense in C for every r > 0, where Dr is the disc centered at the origin with radius r. In particular, this

−1 means that f (Dr) is unbounded, and hence noncompact. This contradicts the properness

of f. Therefore, if f is a proper holomorphic map C → C, then the power series expansion of f must be finite, and hence, f must be a polynomial.

Example 3. In contrast with the previous two examples, the only proper holomorphic self-

n mappings of the unit ball Bn = {z ∈ C : kzk < 1}, n ≥ 2, are the automorphisms of Bn. This result, first proven by Alexander [5], the details of which can also be found in [25, page 316], shows that on some domains, the notions of biholomorphism and of proper holomorphic maps are actually equivalent. 11 For future reference, each automorphism of the unit ball Bn is of the form:

a − P z − s Q z ϕ (z) = a a a , (I.2) a 1 − hz, ai

n where a ∈ Bn, Pa is the orthogonal projection from C onto the one dimensional complex 1 n linear subspace spanned by a, Qa = − Pa is the orthogonal projection from C onto the orthogonal complement of the one dimensional complex linear subspace spanned by a, and

2 1 sa = (1 − |a| ) 2 . A proof of this fact can be found in [25, page 25].

12 CHAPTER II ANALYTIC SETS AND PROPER MAPS

II.1. Analytic Subsets of Complex Manifolds We assume the basic definitions and facts regarding real and complex manifolds, analytic sets and their mappings. Complete treatments of these topics may be found in the texts [9], [21], and [26]. However, for completeness we collect here some definitions and facts which will be needed in this work. We also provide proofs of some results which are needed in the sequel, but whose proofs cannot be found in the standard texts mentioned above.

Definition 2. The following definitions will be important in our work:

1.A complex manifold is a Hausdorff space M with a collection of maps Φα :

n Uα → C , where each Uα ⊂ M is open, such that [ (a) M = Uα, α (b) each Φα is a homeomorphism Uα → Φα(Uα), and

−1 (c) for each pair of maps Φα and Φβ,Φα ◦ Φβ is holomorphic on Φβ(Uα ∩ Uβ).

The collection of maps {Φα} is called a chart. We call n the dimension of the manifold.

2. Let M and N be complex manifolds with charts {Φα} and {Ψβ}. A map

−1 f : M → N is a holomorphic map if, for every Φα and Ψβ, the composition Ψβ ◦f ◦Φα

−1 is holomorphic on Φα(f (Uβ) ∩ Uα). This allows us to define holomorphic maps (C- valued maps) on any open subset of M 3. Let M be a complex manifold. A set A ⊂ M is called an analytic subset of

M if for each point a ∈ M there is a neighborhood U of a and functions f1, . . . , fn holomorphic on U such that

A ∩ U = {z ∈ U : f1(z) = f2(z) = ··· = fn(z) = 0}.

If A is an analytic subset of an open subset Ω ⊂ M, we call A a local analytic subset of M. 13 4. Let A be a local analytic subset of a complex manifold M. A function f : A → C is holomorphic if for each a ∈ A, there is an open neighborhood U of a in M and a

holomorphic function F : U → C such that f ≡ F on A ∩ U. 5. Let X and Y be local analytic subsets of complex manifolds M and N respec- tively. A map f : X → Y is a holomorphic map if for each x ∈ X, there is an open neighborhood U of x in M, and a holomorphic function F : U → N such that f ≡ F on X ∩ U. 6. Let A be a local analytic subset of a complex manifold M. A point a ∈ A is regular if there is an open neighborhood U of a in M such that U ∩ A is a submanifold of M. The collection of regular points of A is denoted reg A. The dimension of this

submanifold is called the dimension of A at a, denoted dima A. 7. The dimension of an analytic subset A of a complex manifold M is defined to be

dim A = max {dima A}. a∈reg A

8. An analytic set A is called reducible if there are analytic sets A1 and A2 distinct

from A such that A = A1 ∪ A2. An analytic set that cannot be represented in this manner is called irreducible.

For the following results, we refer to [9] for proofs, whereas Riemann’s Continuation Theorem can be found in [21].

Proposition 7. Let A be an analytic subset of a complex manifold M. Then, we have the following: 1. A is closed in M. 2. If A 6= M, then M \ A is dense in Ω. 3. M \ A is connected.

Theorem 4 (Riemann’s Continuation Theorem.). Let M be a complex manifold and A an analytic subset of M such that M \ A is dense in M. Let f be holomorphic on M \ A and 14 suppose that for any a ∈ A, there exists an open neighborhood U of a in M such that f|U\A is bounded. Then, there is a unique function F , holomorphic on M, such that F |M\A = f.

II.2. Proper Holomorphic Maps The following results, which we will need, and their proofs, can be found in [9].

Proposition 8. 1. Let A ⊂ Cn be a compact local analytic set. Then A is finite. 2. An analytic set A is irreducible if and only if the set reg A is connected. 3. Let A and A0 be analytic subsets of a complex manifold Ω. Then A \ A0 is also an analytic subset in Ω.

The following remarkable theorem of Remmert shows that the proper holomorphic image of an analytic set is analytic. A proof of this can be found in [9].

Theorem 5 (Remmert’s Theorem). Let A be an analytic subset of a complex manifold M and let f : A → Y be a proper holomorphic map into another complex manifold Y . Then f(A) is an analytic subset of Y of dimension dim f.

In the above theorem, dim f is defined to be the maximum codimension of the fibers f −1(x), which is equal to the rank of f at any regular point. The following theorem, which is encompassed by the famous GAGA set of analogies, is a consequence of Remmert’s Theorem. A proof of this can also be found in [9].

Theorem 6 (Chow’s Theorem). Every analytic subset of a complex projective space is an algebraic variety.

n m Proposition 9. Let X and Y be analytic subsets of Ω1 ⊂ C and Ω2 ⊂ C , respectively, and let f : X → Ω2 be a proper holomorphic map such that f(X) ⊂ Y . Then, 1. If f(X) = Y , then dim X = dim Y . 2. If Y is irreducible and dim X = dim Y , then f(X) = Y .

15 Proof. By Remmert’s Theorem, f(X) is an analytic subset of Ω2, with dim f(X) = dim f. Since f is proper, we cannot have dim f < dim X. Otherwise, the Rank Theorem [24, page 229] would imply that for some y ∈ f(X), f −1(y) is a compact analytic subset of X with positive dimension. By Proposition 8, this is impossible. Hence, we conclude that dim f(X) = dim X. 1. Suppose f(X) = Y . Then, evidently, dim X = dim f(X) = dim Y . 2. Now suppose dim X = dim Y and suppose that f(X) 6= Y . Then, by Proposi-

tion 8, Y \ f(X) is an analytic set, and since Y is closed in Ω2, Y \ f(X) is contained in Y . Additionally, since dim f(X) = dim Y , Y \ f(X) cannot be all of Y . Now, since Y = f(X) ∪ Y \ f(X), Y is reducible. Hence, if Y is irreducible and dim X = dim Y , then f(X) = Y .

Note that, in the case that X and Y are complex manifolds, Y is irreducible if and only if Y is connected, which immediately gives us the following corollary:

Corollary 2. Let X and Y be complex manifolds, and let f : X → Y be a proper holomorphic map. Then, 1. If f(X) = Y , then dim X = dim Y . 2. If Y is connected and dim X = dim Y , then f(X) = Y .

In the following definition, Tp(M) denotes the tangent space to M at the point p ∈ M.

It’s important to note that the dimension of Tp(M) at every point p ∈ M is equal to the dimension of the manifold M.

Definition 3. Let f : X → Y be a smooth map of manifolds and Z a submanifold of Y .

−1 We say the map f is transversal to Z if Image(dfx) + Tf(x)(Z) = Tf(x)(Y ) for all x ∈ f (Z).

The following theorem, and its corrollary that we use to prove Proposition 10, as well as their proofs, can be found in [17]. 16 Theorem 7 (The Transversality Theorem). Suppose that F : X × S → Y is a smooth map of manifolds, where only X has boundary, and let Z be any boundaryless submanifold of Y .

If both F and ∂F are transversal to Z, then for almost every s ∈ S, both fs and ∂fs are transversal to Z.

Note that in the above theorem, ∂F denotes the restriction of F to the boundary

∂(X × S), fs : X → Y is defined by fs(x) = f(x, s), and hence ∂fs is the restriction of fs to ∂X.

Corollary 3. Suppose f : X → Y is a smooth map of manifolds, where only X has boundary, and the boundary map ∂f : ∂X → Y is transversal to Z, then there is a map g : X → Y homotopic to f with ∂g = ∂f, which is transversal to Z.

The proposition below can be found in [16]. Although [16] uses a different technique, the proof we present below is based on the Transversality Theorem.

Proposition 10. Let M be a smooth, connected real manifold without boundary, A ⊂ M be closed submanifold, x a point in M \ A, and i : M \ A → M the inclusion map. Then 1. if the codimension of A is at least 2, the group homomorphism

i∗ : π1(M \ A, x) → π1(M, x)

is surjective, and 2. if the codimension of A is at least 3, then the group homomorphism

i∗ : π1(M \ A, x) → π1(M, x)

is an isomorphism.

Proof. 1. Let x0 ∈ M \ A and f : [0, 1] → M be a smooth, closed path starting

at x0. Since x0 6∈ A, ∂f is transversal to A. By Corollary 3, there is a smooth path

g : [0, 1] → M starting at x0, which is homotopic to f and is transversal to A. By the 17 transversality condition of Definition 3, we have Image(dgx) + Tg(x)(A) = Tg(x)(M) for all x ∈ g−1(A). Now, since g([0, 1]) is a one-dimensional submanifold of M, if the codimension of A is at least 2, the only way that g can satisfy the transversality condition is if g([0, 1])∩A = ∅.

Since every closed path in M starting at x0 is homotopic to a smooth path in M

starting at the same point, this proves that for every [f] ∈ π1(M, x0), there is some

[g] ∈ π1(M \ A, x0) with f ' g. Hence, the map i∗ : π1(M \ A, x) → π1(M, x) is surjective. 2. Let h : [0, 1] × [0, 1] → M be a homotopy of closed paths in M \ A starting at

x0. Assume that h is smooth. Since the paths h([0, 1] × {0}) and h([0, 1] × {1}) are

in M \ A, and x0 ∈ M \ A, ∂h is transversal to A. By Corollary 3, there is a smooth map g : [0, 1] × [0, 1] → X with ∂g = ∂h, which is homotopic to h and is transversal to A. Since ∂g = ∂h, g is a homotopy of the same paths that h is. By the transversality

−1 condition of Definition 3, we have Image(dgx)+Tg(x)(A) = Tg(x)(M) for all x ∈ g (A). Now, since g([0, 1] × [0, 1]) is a two-dimensional submanifold of M, if the codimension of A is at least 3, the only way that g can satisfy the transversality condition is if

g([0, 1] × [0, 1]) ∩ A = ∅. Since every closed path in M \ A starting at x0 is homotopic to a smooth path starting at the same point, and every homotopy of smooth paths

is homotopic to a smooth homotopy, this proves that for every pair of paths f1 and

f2 in M \ A with f1 ' f2 in M, we also have f1 ' f2 in M \ A. Hence, the map

i∗ : π1(M \ A, x) → π1(M, x) is injective.

The following proposition regarding the structure of analytic sets is called the strati- fication of analytic sets. It tells us, in particular, that each analytic set can be decomposed into a disjoint union of manifolds, and that for each integer p, 1 ≤ p ≤ dim A, the union of the manifolds with dimension at most p is an analytic set. More details can be found in [9].

18 We use this stratification of analytic sets to show that Proposition 10 can be generalized to the case where A is an analytic set, rather than just a closed submanifold. Let t denote the disjoint union, that is, a union of disjoint sets. Then we have the following:

Proposition 11. Let A be an analytic subset of a complex manifold M. Then, there exist

disjoint local analytic subsets Aj of A such that

n G A = Ai, i=1

Fk where Bk = i=1 Ai is an analytic subset of M and Ak is a closed submanifold of M \ Bk−1 for each 1 ≤ k ≤ n.

Proposition 12. Let M be a smooth, connected complex manifold without boundary, A ⊂ M be an analytic subset, x a point in M \ A, and i : M \ A → M the inclusion map. Then 1. if the complex codimension of A is at least 1, the group homomorphism

i∗ : π1(M \ A, x) → π1(M, x)

is surjective, and 2. if the complex codimension of A is at least 2, then the group homomorphism

i∗ : π1(M \ A, x) → π1(M, x)

is an isomorphism.

Proof. By Proposition 11, we can write

n G A = Ai, i=1

Fk where Bk = i=1 Ai is an analytic subset of M, hence closed in M, and Ak is a closed

submanifold of M \ Bk−1 for each 1 ≤ k ≤ n. Note that, if A has codimension at least s in

M, then Ak also has codimension at least s in M \ Bk−1 for each 1 ≤ k ≤ n. Thus, assuming

19 A 6= M,

Mk = M \ Bk

is a open submanifold of M for each 1 ≤ k ≤ n. Hence, since

Mk = Mk−1 \ Ak, each inclusion in the following chain i i i i M \ A = Mn Mn−1 ... M1 M0 = M

satisfies the conditions of Proposition 10. Thus, it follows that if the complex codimension

of A is at least 1, i∗ : π1(Mk, x) → π1(Mk−1, x) is surjective for each 1 ≤ k ≤ n, and if the

complex codimension of A is at least 2, i∗ : π1(Mk, x) → π1(Mk−1, x) is an isomorphism for each 1 ≤ k ≤ n. The conclusion follows.

20 CHAPTER III SYMMETRIC POWERS

III.1. Definitions and Preliminaries In the definitions below and throughout the paper, recall that we use superscripts to denote elements in a symmetric product. Usually the topological space will be Cs, and we

i th i s use xj to denote the j component of x ∈ C . Let X be a set, and for a positive integer m, let Xm denote the m-th Cartesian power of X, which is by definition the collection of ordered tuples

{(x1, . . . , xm), xj ∈ X, j = 1, . . . , m}.

m The symmetric group Sm of bijective mappings of the set {1, . . . , m} acts on X in a natural

1 m m way: for σ ∈ Sm, and x = (x , . . . , x ) ∈ X we set

σ · x = (xσ(1), . . . , xσ(m)).

m m Definition 4. The m-th Symmetric Power of X, denoted by XSym is the quotient of X

m under the action of Sm defined above, i.e, points of XSym are orbits of the action of Sm on Xm. We denote by

m m π : X → XSym the natural quotient map.

1 m m m If x = (x , . . . , x ) ∈ X , we denote its image in XSym by

π(x) = hx1, . . . , xmi,

m so that we can think of π(x) as an unordered m-tuple of points of X. For elements of XSym we use the notation in [26]: we denote by

1 2 k hx : m1, x : m2, . . . , x : mki (III.1)

21 Pk where m1, m2, . . . , mk are non-negative integers (called multiplicities) with j=1 mj = m,

m j the element of XSym in which x is repeated mj times. Also notice, that the construction of the symmetric power is functorial, i.e., given a

m m m mapping between sets f : X → Y , there is a natural map fSym : XSym → YSym, called the symmetric power of the map f, defined by

m 1 m
1 m fSym hx , . . . , x i = hf(x ), . . . , f(x )i. (III.2)

m It is not difficult to see that fSym is well defined. Further, it is easy to verify that the

m m m m construction is functorial, in the sense that (id ) = id m and (f ◦ g) = f ◦ g . X Sym XSym Sym Sym Sym

III.2. Topological Symmetric Powers The notion of symmetric power makes sense in a category where one can define finite products of objects and quotients of objects by finite groups. For example, one can construct the n-fold symmetric power of any topological space, which will also be a topological space in a natural way. Let X be a locally compact Hausdorff topological space and let m be a positive

m integer. We endow XSym with the quotient topology. Recall that in this topology a subset

m −1 m U of XSym is open if and only if π (U) is open in X . We now consider two properties of the map π.

m m Proposition 13. The map π : X → XSym is a proper map.

m S −1 Proof. Let K ⊂ XSym be compact, and let α∈A Oα be an open cover of π (K). We refine

−1 the cover as follows: For every x ∈ K, each element xe in the fiber π (x) is contained in

Oα for some αx ∈ A. Additionally, each x has an open neighborhood Ux ⊂ Oα , and these xe e e e xe neighborhoods can be made to be disjoint. Set

\ V = π(U ). x xe −1 xe∈π (x)

22 −1 Then, π (Vx) is a disjoint union of open neighborhoods Vx, with x ∈ Vx ⊂ Ux ⊂ Oα . Then, e e e e xe

[ [ V xe −1 x∈K xe∈π (x) S is a refinement of the cover α∈A Oα. Now, since x ∈ Vx, and since π is a quotient map,

[ Vx x∈K is an open cover of K. Since K is compact, there is a finite set F ⊂ K, where

[ Vx x∈F

−1 −1 still covers K. Now, if y ∈ Vx, we certainly have π (y) ⊂ π (Vx). Hence,

[ [ V xe −1 x∈F xe∈π (x) still covers π−1(K). Since each fiber π−1(x) contains at most m! elements, this is a finite

−1 S cover. Now, since π (K) can be covered by a finite refinement of α∈A Oα, we know that S −1 α∈A Oα must have a finite subcover. Hence, π (K) is compact, and so π is proper.

Remark. In the above proposition, we only used two properties of the map π: 1. π is a quotient map of topological spaces. 2. Each element in the range of π has at most m! preimages.

In general, if p is a quotient map and for some n ∈ N, each element in the range of p has at most n preimages, then p is proper.

Now let m1, . . . , mk be a partition of m, i.e., mj are positive integers such that

Pk m j=1 mj = m. Let V (m1, . . . , mk) be the set of points α in XSym such that there are distinct x1, . . . , xk ∈ X such that

1 k α = hx : m1, . . . , x : mki,

23 j where we use the notation (III.1), that is x is repeated mj times. Also let

−1 m Ve(m1, . . . , mk) = π (V (m1, . . . , mk)) ⊂ X .

Proposition 14. The restricted map

π : Ve(m1, . . . , mk) → V (m1, . . . , mk) is a covering map of degree m! . m1!m2! ··· mk!

Proof. Let α ∈ V (m1, . . . , mk). We will show that there is an open set Uα ⊂ V (m1, . . . , mk),

−1 i containing α, such that π (Uα) is a disjoint union of open subsets Ueα ⊂ Ve(m1, . . . , mk),

i i with the restriction of π to each Ueα a homeomorphism onto its image π(Ueα).

1 k 1 k We can write α = hx : m1, . . . , x : mki for some distinct x , . . . , x ∈ X. Since X is

i Hausdorff, there exist disjoint open subsets Ui ⊂ X with x ∈ Uj if and only if i = j. Now, let

1 k i Uα = {hα : m1, . . . , α : mki ∈ V (m1, . . . , mk) | α ∈ Ui}.

Note that Uα is open in in the subspace topology defined on V (m1, . . . , mk), since

m1 mk Uα = π(U1 × · · · × Uk ) ∩ V (m1, . . . , mk),

mi 1 m σ(1) σ(m) where Ui denotes the mi-fold Cartesian power of Ui. Let σ(y , . . . , y ) denote (y , . . . , y )

1 m m −1 1 m 1 m for σ ∈ Sm and (y , . . . , y ) ∈ X . Then, π (hy , . . . , y i) = {σ(y , . . . , y ) | σ ∈ Sm}, and we have

  −1 [ m1 mk π (Uα) = σ (U1 × · · · × Uk ) ∩ Ve(m1, . . . , mk) σ∈Sm   [ m1 mk = σ(U1 × · · · × Uk ) ∩ Ve(m1, . . . , mk) σ∈Sm ! [ m1 mk = σ(U1 × · · · × Uk ) ∩ Ve(m1, . . . , mk). σ∈Sm 24 m1 mk m −1 Since the sets σ(U1 × · · · × Uk ) are open in X , it follows that π (Uα) is open in

Ve(ma, . . . , mk) with the subspace topology. Also, since Ui ∩ Uj = ∅ for i 6= j, the sets

m1 mk m1 mk σ(U1 ×· · ·×Uk ) and τ(U1 ×· · ·×Uk ) are either identical or disjoint for each σ, τ ∈ Sm,

m1 mk and the number of distinct sets σ(U1 ×· · ·×Uk ) is equal to the number of distinct preimages

−1 1 k π (hα : m1, . . . , α : mki), which is exactly

m! . m1!m2! ··· mk!

m1 mk m1 mk Now, the restricted map π : σ(U1 × · · · × Uk ) → π(U1 × · · · × Uk ) is one-to-one, and hence a homeomorphism, as π is a quotient map. Thus, by restricting π to the subspace

m1 mk m1 mk σ(U1 × · · · × Uk ) ∩ Ve(m1, . . . , mk), we have π : σ(U1 × · · · × Uk ) ∩ Ve(m1, . . . , mk) → Uα is a homeomorphism as well.

III.3. Algebraic Embedding of Symmetric Powers of Projective Spaces In this section we describe a method of realizing the m-th symmetric power of a vector space as a subset of its m-th symmetric tensor product. In view of our applications, all our vector spaces may be assumed to be complex and finite dimensional. As usual

V1 ⊗ V2 ⊗ · · · ⊗ Vm denotes the tensor product of the m vector spaces V1,V2 ...,Vm. For a detailed treatment of tensor products, see [10]. Then V1 ⊗ V2 ⊗ ...... Vm is a vector space Qm of dimension j=1 dim Vj. The symmetric tensor product of Hilbert spaces arise in the context of systems of identical particles in quantum mechanics. For a two particle system, with one particle in state ψ1 and the other in state ψ2, if these particles are distinguishable, the state of the system is represented by ψ1 ⊗ ψ2. However, when the particles are indistinguishable, the convention is to symmetrize the state, representing it instead by √1 (ψ ⊗ψ +ψ ⊗ψ ). This 2 1 2 2 1 is, up to a constant, the projection of ψ1 ⊗ ψ2 onto the symmetric part of the tensor algebra, and is equivalent to representing the state by ψ1 ψ2.

25 Definition 5. The m-fold symmetric tensor product of a vector space V , denoted V m is defined as V m = V ⊗m/ ∼, where V ⊗m = V ⊗ V ⊗ · · · ⊗ V is the m-fold tensor product of V with itself, and the equivalence relation ∼ is defined by:

vσ(1) ⊗ vσ(2) ⊗ · · · ⊗ vσ(m) ∼ v1 ⊗ v2 ⊗ · · · ⊗ vm (III.3)

1 2 m 1 2 m for all v , v , . . . , v ∈ V and all σ ∈ Sm. We denote the equivalence class of v ⊗v ⊗· · ·⊗v under this equivalence relation (which is an element of V m) by

v1 v2 · · · vm.

Just as the tensor product of vector spaces is itself a vector space, the symmetric tensor product V m of a vector space is again a vector space, with a natural linear structure as a quotient vector space of V ⊗m. Suppose V is finite-dimensional of dimension s + 1,

s+1 and let {e0, . . . , es} be a basis of V . Let µ = (µ0, µ2, . . . , µs) ∈ N be a multi-index with Ps m |µ| = j=0 µj = m, and let eµ denote the element of V given by

eµ = ei1 ei2 · · · eim (III.4)

3 where exactly µj of the eik are equal to ej. For example, if V = C with basis {e0, e1, e2},

3 2 the elements eµ ∈ (C ) are:

e(2,0,0) = e0 e0, e(0,2,0) = e1 e1, e(0,0,2) = e2 e2,

e(1,1,0) = e0 e1, e(1,0,1) = e0 e2, e(0,1,1) = e1 e2.

m It is not difficult to see that {eµ}|µ|=m gives a basis for V . Therefore the dimension

m Ps of V is the number of solutions of j=0 µj = m, i.e.

m + s dim V m = . (III.5) m

m Given a vector space V , there is a natural mapping from the symmetric power, VSym

26 (which is just a set), to the symmetric tensor product V m (which is a vector space), given by hv1, v2, . . . , vmi 7→ v1 v2 · · · vm. (III.6)

This mapping is well-defined, since both the unordered tuple hv1, v2, . . . , vmi and the sym- metric tensor v1 v2 · · · vm are invariant under a permutation of the variables. Let V be a vector space, and denote by P(V ) the projectivization of V , that is, the collection of all one-dimensional linear subspaces of V . In other words, we take the quotient of V \{0}, subject to the equivalence relation

λv ∼ v (III.7)

for all λ ∈ C \{0}, and imbue it with the quotient topology. For a vector v in a vector space V [v] ∈ P(V )

denotes its equivalence class in the projectivization of V . In the case that V = Cs+1, P(V ) is a complex manifold of dimension s. There is a natural map

P m P m ψ :( (V ))Sym → (V )

induced by the map (III.6) given by

ψ(h[v1], [v2],..., [vm]i) = v1 v2 · · · vm ∈ P(V m), (III.8)

with vj ∈ V . We call this the Segre-Whitney map. A detailed treatment of symmetric powers and this mapping can be found in the appendix of [26]. This is a symmetric version of the classical Segre embedding of the product of two or more projective spaces as a projective variety,

P(V1) × P(V2) × · · · × P(Vm) → P(V1 ⊗ V2 ⊗ · · · ⊗ Vm)

27 given by ([v1], [v2],..., [vm]) 7→ [v1 ⊗ v2 ⊗ · · · ⊗ vm].

More details on the Segre map and projective varieties can be found in [18].

Proposition 15. The Segre-Whitney map (III.8) is well-defined and injective.

Proof. First we show that the Segre-Whitney map is well-defined, i.e. it does not depend

j j on the choice of the vectors v but only depends on the elements [v ]. Let λ1, λ2, . . . , λm ∈

C \{0} and let v1, v1, . . . , vm ∈ V . Since both sides of (III.8) are invariant under per-

1 2 m 1 2 m mutation of v , v , . . . , v , we merely need to check that ψ(h[λ1v ], [λ2v ],..., [λmv ]i) = ψ(h[v1], [v2],..., [vm]i):

1 2 m 1 2 m ψ(h[λ1v ], [λ2v ],..., [λmv ]i) = [(λ1v ) (λ2v ) · · · (λmv ]

1 2 m = [λ1λ2 ··· λm(v v · · · v )]

= [v1 v2 · · · vm]

= ψ(h[v1], [v2],..., [vm]i).

Hence, ψ is well-defined. Now to show that ψ is injective, suppose that [u1 u2 · · · um] =

1 2 m [v v · · · v ]. Then, there must be constants λ1, λ2, . . . , λm ∈ C\{0} and some σ ∈ Sm such that

1 2 m σ(1) σ(2) σ(m) u ⊗ u ⊗ · · · ⊗ u = (λ1v ) ⊗ (λ2v ) ⊗ · · · ⊗ (λmv ),

i σ(i) 1 2 m i.e., such that u = λiv for each i ∈ {1, 2, . . . , m}. Hence, if ψ(h[u ], [u ],..., [u ]i) = ψ(h[v1], [v2],..., [vm]i), then

1 2 m σ(1) σ(2) σ(m) h[u ], [u ],..., [u ]i = h[λ1v ], [λ2v ],..., [λmv ])

= h[vσ(1)], [vσ(2)],..., [vσ(m)]i

= h[v1], [v2],..., [vm]i.

This proves injectivity. 28 s+1 When V = C , with coordinates (z0, z1, . . . , zs), the Segre-Whitney map is therefore a map Ps m P s+1 m
∼ PN(m,s) ψ :(C )Sym → (C ) = C , (III.9) where (see (III.5))

m + s N(m, s) = − 1 = dim ( s+1) m
− 1. (III.10) m C C

We will now describe the map ψ as in (III.9) explicitly. To do this, we note that since

s+1 m s+1 m {eµ}|µ|=m is a natural basis of (C ) , the linear coordinates of a point w ∈ (C ) P s+1 m can be written as (wµ)|µ|=m, and the homogeneous coordinates of [w] ∈ ((C ) ) as   (wµ)|µ|=m . Recall that a polynomial p in variables x1, x2, . . . , xm is called homogeneous if each term of p has the same degree. If a polynomial is homogeneous, the degree of each term is the same as the degree of the polynomial.

Proposition 16. For each µ ∈ Ns+1 with |µ| = m, there is a homogeneous polynomial of degree m in (s + 1)m variables

s+1 m ψµ :(C ) → C such that for z1, . . . , zm ∈ Cs+1 we have   1 m X 1 2 m ψ(h[z ],..., [z ]i) =  ψµ(z , z , . . . , z )eµ |µ|=m

Remark: the polynomials ψµ are called multisymmetric.

P s+1 m P s+1 m Proof. This is a computation. Recall that π :( (C )) → ( (C ))Sym is the natural P s+1 m P s+1 m quotient map, and ψ :( (C ))Sym → ((C ) ) is the Segre-Whitney map. The com- position of these two maps ψ ◦ π gives a mapping (P(Cs+1))m → P((Cs+1) m). Denote by

s+1 s+1 e0, e1, . . . , es the standard basis of C . Recall that for µ ∈ N be a multi-index with

29 (s+1) m |µ| = m and indices µ0, µ1, . . . , µs, eµ denotes the element of (C ) given by

eµ = ei1 ei2 · · · eim

1 2 m s+1 where exactly µj of the eik are equal to ej. Let z , z , . . . , z ∈ C . Then

(ψ ◦ π)([z1], [z2],..., [zm]) = [z1 z2 · · · zm]

1 1 1 2 2 2 = [(z0e0 + z1e1 + ··· + zs es) (z0e0 + z1e1 + ··· + zs es) ...

m m m (z0 e0 + z1 e1 + ··· + zs es)]   X 1 2 m =  ψµ(z , z , . . . , z )eµ , (III.11) |µ|=m

where the ψµ are the components of ψ. As we can see from the above equation, for each

|µ| = m, each ψµ has the form

1 2 m X 1 2 m ψµ(z , z , . . . , z ) = zi1 zi2 ··· zim , (III.12)

where the sum ranges over all sequences i1, . . . , im with exactly µj of the ik’s equal to j. These polynomials are precisely the degree m homogeneous multi-symmetric polynomials in z1, z2, . . . , zm.

P s+1 m We now show that the Segre-Whitney map can be used to embed ( (C ))Sym into P((Cs+1) m) as an analytic set.

P s+1 m P s+1 m Proposition 17. Let ψ :( (C ))Sym → ((C ) ) be the Segre-Whitney map. Then, P s+1 m P s+1 m ψ(( (C ))Sym) is a projective algebraic variety in the projective space ((C ) ).

Proof. Since (P(Cs+1))m is a product of compact spaces, it is itself compact, and since ψ ◦ π is continuous, it is automatically proper. Since from the representation (III.11), the mapping ψ ◦ π is holomorphic, by Remmert’s Theorem (Theorem 5), ψ ◦ π maps the image of (P(Cs+1))m in P((Cs+1) m) as an analytic set. Furthermore, since (ψ ◦ π)((P(Cs+1))m) =

30 P s+1 m ψ(( (C ))Sym) is an analytic subset of a projective space, by Chow’s Theorem (Theorem 6), it is actually a projective algebraic variety.

III.4. Embedding of Symmetric Powers of Affine Spaces and of Balls

Let E be a subset of CPs, and let i : E → CPs be the inclusion map. Then we have a natural inclusion m m Ps m iSym : ESym → (C )Sym,

m Ps m so that we can think of ESym as a subset of (C )Sym. Composing with the Segre-Whitney Ps m PN(m,s) embedding ψ of (C )Sym in C , where N(m, s) is defined as in (III.10), we obtain an embedding m m PN(m,s) ψ ◦ iSym : ESym → C which we will again call the Segre-Whitney embedding. In this way we can think of symmetric powers as sitting in some projective space. We will denote this embedded version of the m-th symmetric power of E by ΣmE, i.e.,

m m m PN(m,s) Σ E = ψ ◦ iSym(ESym) ⊂ C . (III.13)

s+1 Let e0, . . . , es be the natural basis of C and let z0, . . . , zs be the corresponding linear coordinates, which we can think of as homogeneous coordinates on the projective

Ps Ps space C . We consider the affine piece A0 of C defined by

 Ps A0 = [z0 : ··· : zs] ∈ C | z0 6= 0 .

s s We can identify A0 with C in a natural way, by mapping a point (z1, . . . , zs) ∈ C to the Ps s+1 point of A0 ⊂ C given by [1 : z1 : ··· : zs]. Recall that for each multi-index µ ∈ N such that |µ| = m, if we define eµ by (III.4), then the collection {eµ}|µ|=m forms a basis of the vector space (Cs+1) m. We denote the linear coordinates with respect to this basis by

31 s+1 s+1 m (zµ)|µ|=m. Let µ0 = (m, 0, 0,... ) ∈ N , so the corresponding basis element of (C ) is

eµ0 = e0 e0 · · · e0.

P s+1 m PN(m,s) Let Aµ0 be the affine piece of (C ) ) = C (see (III.10)) given by

 P s+1 m ∼ PN(m,s) Aµ0 = [(zµ)|µ|=m] ∈ ((C ) )) = C | zµ0 6= 0 .

N(m,s) Again, we can identify Aµ0 with C with coordinates (zµ)µ6=µ0 by mapping the element h i   P z e ∈ A to the point zµ ∈ N(m,s). |µ|=m µ µ µ0 zµ C 0 µ6=µ0 We have the following:

m m Ps m Proposition 18. The mapping ψ ◦ iSym maps the subset (A0)Sym of (C )Sym into the affine P s+1 m ∼ PN(m,s) piece Aµ0 of ((C ) )) = C .

m 1 2 m i Proof. Let z ∈ (A0)Sym. Then, z = h[z ], [z ],..., [z ]i, where each [z ] ∈ A0, and hence j z0 6= 0 for each j. Then,

m 1 2 m (ψ ◦ iSym)(z) = ψ(h[z ], [z ],..., [z ]i)   X 1 2 m =  ψµ(hz , z , . . . , z i)eµ , |µ|=m

1 2 m where the components are as in (III.12). We see that, for µ0 = (m, 0,..., 0), ψµ0 (hz , z , . . . , z i) =

1 2 m m m m m z0z0 ··· z0 6= 0. Hence, for z ∈ (A0)Sym, we have ψ◦iSym(z) ∈ Aµ0 , and so ψ◦iSym((A0)Sym) ⊂

Aµ0 .

s Ps Therefore, identifying A0 with C , denoting by i the inclusion of A0 in C and

N(m,s) identifying Aµ0 with C (see (III.10)), we obtain an injective continuous map

m s m N(m,s) φ = ψ ◦ iSym :(C )Sym → C , which we will continue to call the Segre-Whitney embedding. Recall from (III.13) that its image is denoted by ΣmCs. 32 s m m s Proposition 19. The Segre-Whitney embedding φ :(C )Sym → Σ C is a homeomorphism.

Proof. Since i : Cs → i(Cs) ⊂ P(Cs+1) is a homeomorphism, im :(Cs)m → im((Cs)m) ⊂ P s+1 m m s m m s m P s+1 m ( (C )) , and hence iSym :(C )Sym → iSym((C )Sym) ⊂ ( (C ))Sym are also homeomor- phisms. Since ψ ◦ π is a proper map (see proof of Proposition 17), where π is the quotient

P s+1 m P s+1 m map ( (C )) → ( (C ))Sym, by Proposition 5, ψ is a proper map as well. Since ψ is P s+1 m P s+1 m also injective, by Proposition 6, ψ is a homeomorphism ( (C ))Sym → ψ(( (C ))Sym) ⊂ P s+1 m m s m m s ((C ) ). Thus, it follows that φ = ψ ◦ iSym is a homeomorphism (C )Sym → Σ C .

m s+1 Proposition 20. Let ψ be the Segre-Whitney map, φ = ψ ◦ iSym, µ0 = (m, 0,..., 0) ∈ N , P s+1 m and let Aµ0 be the affine subspace of ((C ) ) given by zµ0 6= 0, i.e.,     X  N(m,s) Aµ0 =  zµeµ : zµ0 6= 0 = C ,  |µ|=m 

m s P s+1 m Where N(m, s) is as defined in (III.10).Then, Σ C = ψ(( (C ))Sym) ∩ Aµ0 , and hence,

s m N(m,s) φ embeds (C )Sym into C as an affine algebraic variety, where N(m, s) is as defined in (III.10).

m s P s+1 m Proof. As we have seen above, Σ C ⊂ ψ(( (C ))Sym) ∩ Aµ0 . All that remains is to show h i s m P s+1 m P that the map φ :(C )Sym → ψ(( (C ))Sym) ∩ Aµ0 is surjective. Let z = |µ|=m zµeµ ∈ P s+1 m ψ(( (C ))Sym)∩Aµ0 . Then, we have zµ0 6= 0, and since z is an element of projective space, P s+1 m without loss of generality, we can assume zµ0 = 1. Now, since z ∈ ψ(( (C ))Sym), we have z = ψ(h[ω1], [ω2],..., [ωm]i) for some ωk ∈ Cs+1, 1 ≤ k ≤ m. Now, since   1 2 m X 1 2 m ψ(h[ω ], [ω ],..., [ω ]i) =  ψµ(hω , ω , . . . , ω i)eµ |µ|=m   X =  zµeµ , |µ|=m

1 2 m 1 2 m k we have ψµ0 (hω , ω , . . . , ω i) = ω0ω0 ··· ω0 6= 0. Now, since ψ is a function of the [ω ],

k 1 2 m we can assume that ω0 = 1 for all 1 ≤ k ≤ m. But this means that h[ω ], [ω ],..., [ω ]i ∈ 33 m s m m s iSym((C )Sym), and hence z ∈ Σ C . m s P s+1 m Since Σ C is the intersection of the projective algebraic variety ψ( (C )Sym) in P s+1 m m s ((C ) ) with the affine plane Aµ0 ,Σ C is an affine algebraic variety, as well as an

analytic subset of Aµ0 .

s m m Proposition 21. Let U ⊂ C be open. Then, Σ U = φ(USym) is a local analytic subset of

N(m,s) the affine space Aµ0 = C (see (III.10)).

s m s m s m m s Proof. Since U is open in C , USym is open in (C )Sym, and since φ :(C )Sym → Σ C is a homeomorphism, we know that ΣmU is an open subset of ΣmCs. Now, we know that ΣmCs

N(m,s) is an analytic subset of Aµ0 = C , and since an open subset of an analytic set is a local analytic set, ΣmU is a local analytic subset of CN(m,s).

What we have shown in this section is that, for an open subset U ⊂ Cs, the abstract

th m m N(m,s) m symmetric power of U, USym, can be realized as a local analytic set Σ U in C . For

m m a complex manifold M and a function f : M → USym or f : USym → M, we can then say that f is holomorphic if f is a holomorphic mapping of analytic sets as in Definition 2.

III.5. Complex Symmetric Powers Symmetric powers of the complex plane and other domains occur naturally in math-

ematics, and also are important in applications. Consider a polynomial p(z) ∈ C[z]. If the degree of p is m, the polynomial p has exactly m zeros in the complex plane, counting multiplicity. Denote these zeros α1, α2, . . . , αm. Then this collection of zeros is well-defined

for every polynomial p in C[z], but has no natural ordering. Thus, it is natural to associate the collection of zeros of p with the unordered tuple hα1, α2, . . . , αmi. This gives a natural

association of the roots of a polynomial of degree m with coefficients in C with elements of

m the space CSym. We compute the Segre embedding ΣmU explicitly for a few simple cases with U ⊂ Cs:

34 2 2 2 P 2 2 1 2 Example 4. In the case of CSym, we have φ : CSym → C ⊂ ((C ) ). Let z , z ∈ C. Then

2 1 2 1 2 ψ ◦ iSym(hz , z i) = [(e0 + z e1) (e0 + z e1)]

1 2 1 2 = [e0 e0 + z e1 e0 + z e0 e1 + z z e1 e1]

1 2 1 2 = [e(2,0) + (z + z )e(1,1) + z z e(0,2)].

2 2 It follows therefore that the Segre-Whitney embedding of CSym in C is given by

1 2 1 2 1 2 2 φ(hz , z i) = (z + z , z z ) ∈ C , (III.14) which is traditionally known as the symmetrization map. Furthermore, φ is surjective onto

C2. Indeed, given a point (s, p) ∈ C2, if z1, z2 are the roots of the equation

t2 − st + p = 0, then we have φ(hz1, z2i) = (s, p). It follows that we can identify

2 2 Σ C = C .

Therefore, we have that

2 1 2 1 2 1 2 Σ D = {(z + z , z z ) | z , z ∈ D}, so that we have recovered the Symmetrized bidisc.

Example 5. We now generalize the previous example from 2 to m variables. In the case of

m m m P 2 m 1 2 m CSym, we have φ : CSym → C ⊂ ((C ) ). Let z , z , . . . , z ∈ C. Then

m 1 2 m 1 2 m ψ ◦ iSym(hz , z , . . . , z i) = [(e0 + z e1) (e0 + z e1) · · · (e0 + z e1)]

1 2 m 1 2 m = [e(m,0) + (z + z + ··· + z )e(m−1,1) + ··· + z z ··· z e(0,m)]

35 It follows that in affine coordinates, the map φ can be represented as

1 2 m 1 m 1 m φ(hz , z , . . . , z i) = (σ1(z , . . . , z ), . . . , σm(z , . . . , z )),

where σk denotes the k-th elementary symmetric polynomial in m variables. Then φ is

m m m again a surjective map from C to C , since given s = (s1, s2, . . . , sm) ∈ C , we can take hz1, . . . , zmi to be the unordered collection of roots of

m m−1 m−2 m t − s1t + s2t − ... (−1) sm = 0,

1 2 m and it follows that φ(hz , z , . . . , z i) = (s1, s2, . . . , sm). Hence, we can identify

m m Σ C = C , and we have

m 1 m 1 m 1 m Σ D = {(σ1(z , . . . , z ), . . . , σm(z , . . . , z )) | z , . . . , z ∈ D}.

Hence, we have also recovered the symmetrized polydisc.

2 2 2 2 5 P 3 2 1 2 2 Example 6. In the case of (C )Sym, we have φ :(C )Sym → C ⊂ ((C ) ). Let z , z ∈ C . Then

2 1 2 1 1 2 2 ψ ◦ iSym(hz , z i) = [(e0 + z1e1 + z2e2) (e0 + z1e1 + z2e2)]

2 2 1 1 2 1 2 = [e0 e0 + z1e0 e1 + z2e0 e2 + z1e1 e0 + z1z1e1 e1 + z1z2e1 e2+

1 1 2 1 2 z2e2 e0 + z2z1e2 e1 + z2z2e2 e2]

1 2 1 2 1 2 1 2 = [e(2,0,0) + (z1 + z1)e(1,1,0) + (z2 + z2)e(1,0,1) + z1z1e(0,2,0) + z2z2e(0,0,2)+

1 2 1 2 (z1z2 + z2z1)e(0,1,1)].

In affine coordinates, we can then represent the map φ as

1 2 1 2 1 2 1 2 1 2 1 2 1 2 φ(hz , z i) = (z1 + z1, z2 + z2, z1z1, z2z2, z1z2 + z2z1).

36 2 2 5 2 2 Note that this map φ :(C )Sym → C is not surjective, but rather the image Σ C is an

5 5 affine algebraic hypersurface in C . If we represent coordinates in C by (x1, x2, x3, x4, x5),

1 2 1 2 1 2 1 2 1 2 1 2 then the map φ is given by x1 = z1 + z1, x2 = z2 + z2, x3 = z1z1, x4 = z2z2, x5 = z1z2 + z2z1, and the image, Σ2C2, is defined by the vanishing of the polynomial (see [22]):

2 2 2 x2x3 − x1x2x5 + x1x4 + x5 − 4x3x4.

Now, for the unit ball, B2, we have

2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 Σ B2 = {(z1 + z1, z2 + z2, z1z1, z2z2, z1z2 + z2z1)|z , z ∈ B2}, which is an open subset of Σ2C2, and hence a local analytic set in C5.

III.6. Symmetric Powers as Complex Spaces

m Thanks to the above constructions, we have a natural identification of (Bs)Sym with a local analytic set in CN(m,s), where N(m, s) is as defined in (III.10). This allows us to

m do complex analysis on (Bs)Sym since we are now able to define holomorphic functions and maps. This construction generalizes the construction of the symmetrized polydisc for s = 1. There is however, another more general and natural approach to the problem of constructing the symmetric power of a complex manifold. This is through the notion of a Complex (Analytic) Space. Complex spaces are a generalization of the notions of complex manifolds and analytic sets. A complex space X is essentially a complex manifold, except

n that the charts Ψα map open subsets Uα ⊂ X onto local analytic sets in C , not just domains.

We also require that a finite or denumerable collection of the Uα’s cover X. In the same way that the charts Ψα give meaning to a function f :Ω → X or f : X → Ω being holomorphic if X is a manifold, so do they when X is a complex space. Therefore, one can do complex analysis on a complex space, since we can define holomorphic functions and maps. For details and precise definitions see [26, 9]. Thanks to a result of H. Cartan (see [6, 7]), whenever G is a finite group of biholo- 37 morphic automorphisms of the complex space X, the quotient spaces X/G has the natural structure of a complex space, such that the quotient map p : X → X/G is holomorphic. We

m could apply this result to the Cartesian power X = (Bs) , and take G = Sm to be the group of biholomorphic automorphisms of X given for σ ∈ Sm by

σ(z1, z2, . . . , zm) = (zσ(1), zσ(2), . . . , zσ(m)).

m Then the symmetric power (Bs)Sym can be identified with the quotient complex space

m (Bs) /Sm.

m One can show (but we will not do this) that the two complex spaces Σ Bs (which is

m a local analytic set in a complex Euclidean space) and (Bs) /Sm (which is a quotient space defined using Cartan’s theorem stated above) are biholomorphic, so we can identify them,

m and think of them as different representations of the abstract symmetric power (Bs)Sym. This is the approach we will take in the last chapter of this thesis.

38 CHAPTER IV PROPER HOLOMORPHIC SELF-MAPS OF SYMMETRIC POWERS OF BALLS In this chapter we prove the main result of this paper, Theorem 3. For completeness,

m we also give a simplified account of the proof of Theorem 1. Recall that (Bs)Sym is the m-fold symmetric power of the unit ball in Cs. We have shown that this can be identified with a

m certain local analytic set Σ Bs, and therefore one can talk about holomorphic maps and functions on this object. In this chapter we determine the holomorphic proper self-maps of

m (Bs)Sym.

IV.1. Parametrized Analytic Sets

Definition 6. Let Ω be an open set in Cn and let

N g :Ω → C

be a holomorphic map. We call either g or g(Ω) a parametrized analytic set. When the dimension n of the source is 1, we call g an analytic curve. We say that g is non-trivial if it is not constant.

Proposition 22. (1) There are no non-trivial parametrized analytic sets in ∂Bs. (2) Let Ω be a connected open set in Cn and let g be a holomorphic function which

m m 1 2 m j takes Ω to the boundary ∂((Bs) ) of (Bs) . If g = (g , g , . . . , g ), where each g maps Ω to

j Bs, then there is at least one j for which g is constant with values in ∂Bs.

n Proof. (1) Let Ω be an open subset of C and g :Ω → ∂Bs be a holomorphic map. Then, we have s X 2 |gi| ≡ 1 i=1 on Ω. Applying the operator ∂2 to both sides of the above equation, we obtain ∂zj ∂zj

s s ∂2 X X ∂2 |g |2 = g g ∂z ∂z i ∂z ∂z i i j j i=1 i=1 j j 39 s X ∂gi ∂gi = ∂z ∂z i=1 j j s X ∂gi ∂gi = ∂z ∂z i=1 j j s 2 X ∂gi = ≡ 0, ∂z i=1 j on Ω for all 1 ≤ j ≤ n. Thus, every first-order partial derivative of g is identically zero on Ω. Hence, g is constant on each connected component of Ω.

m m (2) The boundary ∂(Bs) is a disjoint union of 2 − 1 sets of the form

W1 × · · · × Wk,

where each Wj is either Bs or ∂Bs, and there is at least one Wj which is ∂Bs. Then the union

−1 of the inverse images g (W1 × · · · × Wk) is Ω, and therefore at least one of these inverse images has an interior point. Therefore, there is an open subset ω of Ω, and a j in the set of

j j indices {1, . . . , k} such that g (ω) ⊂ ∂Bs. By part (1), g is a constant on ω, and by analytic continuation, since Ω is connected, gj is constant on Ω also.

The above proposition can be generalized: (1) holds for strongly pseudoconvex do-

s mains U ⊂ C , and hence, (2) holds for products of strongly pseudoconvex domains U1 ×

ni ...Un, Ui ⊂ C . For a proof of the more general version of (1), see [13].

IV.2. Some Results of Remmert-Stein Theory In this section we prove two results, using the powerful technique introduced by Remmert and Stein in [23]. The first result, which is actually found in [23], is the simplest counterexample to a Riemann Mapping Theorem for higher dimensions. In fact it is more general: it shows that there is not even a proper holomorphic map from the bidisc to the ball.

2 Theorem 8. There is no proper holomorphic map D → B2.

40 2 Proof. Suppose to the contrary that there is a proper holomorphic map f : D → B2. Let

∞ f1 and f2 denote the components of f, and let {ζnu}ν=1 ⊂ D be a sequence converging to

∞ some point ζ ∈ ∂D. Then, {fν}ν=1 where fν(τ) = f(ζν, τ) is a sequence of holomorphic

∞ ∞ maps D → D. By Montel’s Theorem, there is a subsequence {ζνn }n=1 for which {fνn }n=1 converges uniformly on compact subsets of D to a holomorphic function g. By Proposition

2, g : D → ∂B2, and by Proposition 22, g, is constant. Hence, by Weierstrass’ Theorem, we have ∂f (ζ , τ) dg i νn → i = 0 ∂τ dτ for i ∈ {1, 2}, as n → ∞.

Thus, since this holds for every sequence {ζν} converging to a point ζ ∈ ∂D, we have

∂f (ω, τ) i → 0 ∂τ for i ∈ {1, 2}, as ω → ∂D. Hence, by the maximum principle,

∂f (ω, τ) i ≡ 0 ∂τ

2 on D . Hence, f is independent of τ, and thus the preimage of a point in B2 must be of the form Ω × D,Ω ⊂ D. However, since D is not compact, Ω × D cannot be compact. Hence, f is not proper.

The following result provides us with the structure of proper holomorphic maps be- tween products of balls. While the main ideas of the proof can be found in [21, page 76], and are ultimately based on ideas of [23], since it does not appear in the literature in this form, we give details of the argument for completeness. The proof uses elementary properties of plurisubharmonic functions, which can be found in [21].

m m Theorem 9. Let s, m be positive integers and let f :(Bs) → (Bs) be a proper holomorphic

1 2 m σ(1) σ(2) σ(m) map. Then, f(τ , τ , . . . , τ ) = (f1(τ ), f2(τ ), . . . , fm(τ )), where each fi is a proper holomorphic self-mapping of Bs and σ is a permutation of {1, 2, . . . , m}. 41 m m Proof. Let f :(Bs) → (Bs) be a proper holomorphic map with components f1, f2, . . . , fm,

∞ 1 2 m and let {ζν}ν=1 ⊂ Bs be a sequence converging to some point ζ ∈ ∂Bs. Let (τ , τ , . . . , τ ) ∈

m (j) 1 j−1 j+1 m (Bs) , and denote by τ the (m−1)-tuple (τ , . . . , τ , τ , . . . , τ ). Fix j ∈ {1, 2, . . . , m},

j (j) 1 j−1 j+1 m j ∞ and define fν (τ ) = f(τ , . . . , τ , ζν, τ , . . . , τ ). Then, {fν }ν=1 is a sequence of holo-

m−1 m ∞ morphic maps (Bs) → (Bs) . By Montel’s Theorem, there exists a subsequence {ζνn }n=1

j ∞ m−1 for which {fνn,}n=1 converges uniformly on compact subsets of (Bs) to a holomorphic

m−1 m function g. Now, by Proposition 2, g :(Bs) → ∂((Bs) ). By Proposition 22, this implies

m−1 that some component of g, say gi is constant on (Bs) . Hence, by Weierstrass’ Theorem, we have 1 j−1 j+1 m (j) ∂fi(τ , . . . , τ , ζνn , τ , . . . , τ ) ∂gi(τ ) k → k = 0 ∂τl ∂τl as n → ∞ for all 1 ≤ k ≤ m, k 6= j and all 1 ≤ l ≤ s.

Thus, for every sequence {ζν} converging to a point ζ ∈ ∂Bs, we have a subsequence

{ζνn } for which

m s 1 j−1 j+1 m 2 X X ∂fi(τ , . . . , τ , ζνn , τ , . . . , τ ) → 0 ∂τ k k=1;k6=j l=1 l

as n → ∞ for some i, which may depend on the sequence {ζν} and the subsequence {ζνn }. However, if we consider the product over i, 1 ≤ i ≤ m, we have

s m s 1 j−1 j+1 m 2 Y X X ∂fi(τ , . . . , τ , ζνn , τ , . . . , τ ) → 0 (IV.1) ∂τ k i=1 k=1;k6=j l=1 l as n → ∞ independently of the choice of ζ ∈ ∂Bs and choice of sequence ζn → ζ. Let m s 2 X X ∂fi hi = . (IV.2) ∂τ k k=1;k6=j l=1 l

Then, not only is hi plurisubharmonic, but log(hi) is plurisubharmonic as well. Hence,

m m ! X Y log(hi) = log hi i=1 i=1

42 is plurisubharmonic, and so is m Qm log( hi) Y e i=1 = hi, i=1 which is precisely the expression in (IV.1). Thus, by the maximum principle, we must have

Qm m i=1 hi ≡ 0 on (Bs) . Thus, since

m m [ m (Bs) = {τ ∈ (Bs) : hi(τ) = 0}, i=1 one of these sets must contain an interior point, and hence a neighborhood, on which hi ≡ 0

m for some i. By analytic continuation, hi ≡ 0 on (Bs) . Thus, by (IV.2), fi depends only on τ j. Furthermore, for each j, the choice of i is uniquely determined, and each i is distinct. For, if there were a choice of i satisfying

m s 1 j−1 j+1 m 2 X X ∂fi(τ , . . . , τ , ζνn , τ , . . . , τ ) ≡ 0 ∂τ k k=1;k6=j l=1 l

0 for two indices j 6= j, then fi would be constant, and f would not be surjective. This is impossible, as, by Corollary 2 f must be surjective. Thus,

f(τ1, τ2, . . . , τm) = (f1(τσ(1)), f2(τσ(2)), . . . , fm(τσ(m))),

where σ is a permutation of {1, 2, . . . , m}. It remains to show that fi(τσ(i)) defines a proper

mapping Bs → Bs for each i ∈ {1, 2, . . . , m}.

m Let K ⊂ Bs be compact. Then, K is compact, and since f is proper,

−1 m −1 −1 −1 f (K ) = fσ−1(1)(K) × fσ−1(2)(K) × · · · × fσ−1(m)(K) (IV.3)

−1 is compact. Hence, we must have fσ−1(i)(K) compact for each i ∈ {1, 2, . . . , m}. Thus, each

fi is a proper holomorphic self-mapping of Bs.

43 IV.3. Results of Edigarian and Zwonek In this section, we give a simplified proof of Theorem 1 following the original argument in [14, 15]. For simplicity, we consider only the case m = 2. The general case differs from this one only in algebraic complexity. Theorem 1 for the case m = 2 will follow from the following result:

2 2 Theorem 10. Let f : D → DSym be a proper holomorphic map. Then, f(ζ, τ) = hB1(ζ),B2(τ)i, where B1,B2 : D → D are proper holomorphic maps.

2 2 2 2 Proof. Let f : D → DSym be a proper holomorphic map, and let φ : DSym → Σ D be the Segre embedding as in (III.14) defined by φ(hα, βi) = (α + β, αβ). Note that hα, βi is

2 precisely the root system of the polynomial p(z) = z − φ1(hα, βi)z + φ2(hα, βi).

Let {ζn} ⊂ D be a sequence converging to some point ζ0 ∈ ∂D. Then, {f(ζn, τ)} is a sequence of holomorphic maps D → D2. By Montel’s Theorem, there exists a subsequence

{ζnk } for which {f(ζnk , τ)} converges uniformly on compact subsets of D to a holomorphic

2 function g(τ). By Proposition 2, we know g : D → ∂(DSym).

2 2 Let Ω1 = ∂(DSym ∩V (1, 1)) and Ω2 = ∂(DSym ∩V (2)), where the notation V (1, 1) and V (2) is as defined in Section III.2. Now, since

−1 −1 D = g (Ω1) t g (Ω2),

−1 where t represents the disjoint union, g (Ωi) must have an interior point for some i ∈

−1 2 2 {1, 2}, and hence an open neighborhood U ⊂ g (Ωi). Since π : D → DSym restricts to a holomorphic covering on each of V (1, 1) and V (2), there is some admissible neighborhood

2 V ⊂ U, on which g lifts to a holomorphic function ge : V → ∂(D ) such that π ◦ ge = g on V . Now, by Proposition 22, one of the components of ge is constant on V with value C ∈ ∂D. Now we consider the polynomial

2 2 z − (φ1 ◦ π ◦ ge)(τ)z + (φ1 ◦ π ◦ ge)(τ) = z − (φ1 ◦ g)(τ)z + (φ2 ◦ g)(τ),

44 where τ ∈ V . Then, evidently, C is a root of this polynomial for each τ ∈ V , and hence

2 C − (φ1 ◦ g)(τ)C + (φ2 ◦ g)(τ) ≡ 0 on V . Since the mapping φ is a biholomorphism, we can consider f as a map D2 → Σ2D and g as a map D → ∂Σ2D. The equation above then takes the simpler form:

2 C − g1(τ)C + g2(τ) ≡ 0 (IV.4)

Since the left hand side of (IV.4) is holomorphic, by analytic continuation, (IV.4) holds on all of D. Differentiating the above equation yields:

dg dg − 1 C + 2 ≡ 0 (IV.5) dτ dτ d2g d2g − 1 C + 2 ≡ 0. (IV.6) dτ 2 dτ 2

Eliminating C from (IV.5) and (IV.6) yields:

d2g dg d2g dg 2 1 − 1 2 ≡ 0. dτ 2 dτ dτ 2 dτ

Now, since f(ζnk , ·) converges uniformly on compact subsets of D to the holomorphic function g, by Weierstrass’ Theorem, we have

∂2f (ζ , τ) ∂f (ζ , τ) ∂2f (ζ , τ) ∂f (ζ , τ) d2g dg d2g dg 2 nk 1 nk − 1 nk 2 nk → 2 1 − 1 2 ≡ 0 ∂τ 2 ∂τ ∂τ 2 ∂τ dτ 2 dτ dτ 2 dτ

as k → ∞. As this is true for any sequence {ζn} → ∂D, by the maximum principle, we have

∂2f ∂f ∂2f ∂f 2 1 − 1 2 ≡ 0 (IV.7) ∂τ 2 ∂τ ∂τ 2 ∂τ

on D2. Hence, defining ∂f2 ∂τ B1 = , (IV.8) ∂f1 ∂τ

∂B1 2  2 ∂f1 by (IV.7), we have ∂τ ≡ 0 on D \A, where A = (ζ, τ) ∈ D | ∂τ (ζ, τ) = 0 is an analytic

45 subset of D2. From (IV.4) and (IV.5), we obtain

dg 2 dg dg dg 2 2 − g 2 1 + g 1 ≡ 0. dτ 1 dτ dτ 2 dτ

By Weierstrass’ Theorem and the Maximum Principle, we obtain a similar equation for f:

∂f 2 ∂f ∂f ∂f 2 2 − f 2 1 + f 1 ≡ 0 (IV.9) dτ 1 ∂τ ∂τ 2 ∂τ on D2. Now, using (IV.8) and (IV.9), we have

2 B1(ζ) − f1(ζ, τ)B1(ζ) + f2(ζ, τ) ≡ 0 (IV.10)

2 2 on D \ A. Note that the above equation implies B1(ζ) is a root of the polynomial z −

2 2 f1(ζ, τ)z + f2(ζ, τ). Since f : D → Σ D, this means that B1(ζ) ∈ D. Hence, B1 is bounded on D2 \ A, and by Riemann’s Continuation Theorem (Theorem 4), extends holomorphically

2 to D. Thus, (IV.10) holds on all of D . Selecting a sequence {τn} → τ0 ∈ ∂D, and repeating the above argument, we can obtain a similar equation:

2 B2(τ) − f1(ζ, τ)B2(τ) + f2(ζ, τ) ≡ 0. (IV.11)

Thus, (IV.10) and (IV.11) tell us that f(ζ, τ) = φ(hB1(ζ),B2(τ)i), and since f is proper, by

Proposition 5, B1,B2 : D → D must be proper holomorphic maps.

Theorem 1 for the case m = 2 now follows from the above result:

2 2 Proof of Theorem 1 (m = 2). Let f : DSym → DSym. Then, f ◦ π is a proper holomorphic

2 2 2 2 mapping D → DSym, where π is the quotient map D → DSym. By the above theorem, f ◦ π has the structure (f ◦ π)(ζ, τ) = hB1(ζ),B2(τ)i, where B1 and B2 are proper holomorphic self-mappings of D. Since these maps must be surjective, there are ζ1, τ1 ∈ D with B1(ζ1) =

B2(τ1) = 0. Since (f ◦ π)(ζ1, τ1) = (f ◦ π)(τ1, ζ1), we evidently have B1(τ1) = B2(ζ1) = 0.

Since this holds for any zeros of B1 and B2, these maps have the same zero set.

Now, let ζ ∈ D such that B1(ζ) 6= 0. As before, since (f ◦ π)(ζ, τ1) = (f ◦ π)(τ1, ζ), 46 we must have B1(ζ) = B2(ζ). Since this holds for any ζ ∈ D, we mush have B1 ≡ B2 on D. The conclusion follows.

The proof of the full version of Theorem 2 can be done in entirely the same manner, but is more complex algebraically. The polynomial (IV.4) is replaced with a polynomial of degree m, and equation (IV.5) is replaced with a system of equations in the partial derivatives

of f, which must be solved in order to define B1 as in (IV.8). For details, see the original paper [15], or [8].

IV.4. Mappings from Cartesian to Symmetric Powers In this section, we begin the proof of Theorem 3, and extend the results of Edigarian

and Zwonek to symmetric powers of the unit ball in Cs for s ≥ 2, which is the main new contribution of this thesis. The first step in the proof of Theorem 3 is the following result, which is interesting in its own right:

m m Theorem 11. Let s ≥ 2, m ≥ 2, and let f :(Bs) → (Bs)Sym be a proper holomorphic map.

m m Then, there exists a proper holomorphic map fe :(Bs) → (Bs) such that f = π ◦ fe.

In other words, the map f can be lifted to a proper holomorphic map fe such that the following diagram commutes:

m (Bs) fe π

m m (Bs) (Bs)Sym f

First, we will need the following lemma:

1 m m i j Lemma 1. Let A = {(z , . . . , z ) ∈ (Bs) : z = z for some i 6= j}. Then π(A) is an

m analytic subset of (Bs)Sym of codimension s. 47 s m ∼ sm Proof. Let Aij be the linear subspace of (C ) = C given by

 1 m i j Aij = (z , . . . , z )|z = z .

i j Then Aij is defined by the vanishing of s linearly independent linear functionals z 7→ zk − zk

s m where 1 ≤ k ≤ s, and consequently Aij is of codimension s in (C ) . Since

[ m A = (Aij ∩ (Bs) ) , i

it now follows that A is an analytic subset of codimension s in (Bs)m. Recall that, by Propo-

m m sition 13, the quotient map π :(Bs) → (Bs)Sym is proper. Since π is also holomorphic, by

m Remmert’s Theorem (Theorem 5), π(A) is an analytic subset of (Bs)Sym. Now, by Proposi-

tion 4, π|A is a proper map A → π(A), and since π is also surjective, by Proposition 9, we

m m must have dim A = dim π(A). Since dim((Bs) ) = dim((Bs)Sym), it follows that π(A) has

m m the same codimension in (Bs)Sym as A has in (Bs) , which is s.

Recall that, for a partition m1, . . . , mk of m, V (m1, . . . , mk) is the set of points α in

m 1 k (Bs)Sym such that there are distinct z , . . . , z ∈ Bs such that

1 k α = hz : m1, . . . , z : mki,

j where this notation is as in (III.1), i.e., z is repeated mj times. Also recall from Section

III.2 the meaning of the notation Ve(m1, . . . , mk):

−1 m Ve(m1, . . . , mk) = π (V (m1, . . . , mk)) ⊂ (Bs) .

s m s m Then, (B ) \ A is precisely the set Ve(m1, . . . , mk) and (B )Sym \ π(A) is precisely the set

V (m1, . . . , mk). We are now ready to prove Theorem 11.

m m Proof of Theorem 11. Let f :(Bs) → (Bs)Sym be a proper holomorphic map and let A be

m m as defined in Lemma 1. Since π(A) is an analytic subset in (Bs)Sym,(Bs)Sym \ π(A) is an 48 open, connected set.

m m Since (Bs) \ A = Ve(1, 1,..., 1) and (Bs)Sym \ π(A) = V (1, 1,..., 1), by Proposition

m m m 14, π|(Bs) \A is a holomorphic covering (Bs) \ A → (Bs)Sym \ π(A). Since a holomorphic m covering is a local biholomorphism, (Bs)Sym \ π(A) is an sm-dimensional complex manifold.

m m Moreover, since π(A) is an analytic subset of (Bs)Sym, by Proposition 7, (Bs)Sym \ π(A) is

m m m connected and dense in (Bs)Sym. Since reg((Bs)Sym) is an open subset of (Bs)Sym containing

m m m (Bs)Sym \ π(A), reg((Bs)Sym) must be connected, and hence by Proposition 8, (Bs)Sym is

m m an irreducible analytic set, with dim((Bs)Sym) = dim((Bs)Sym \ π(A)). Since f is a proper

m m holomorphic map from (Bs) , which is a manifold of dimension sm, to (Bs)Sym, which is an irreducible analytic set of dimension sm, by Theorem 9, f is surjective.

−1 Now, by Proposition 4, f|f −1(π(A)) is a proper holomorphic map from f (π(A)),

m −1 which is an analytic subset of (Bs) , onto π(A), and so by Proposition 9, dim(f (π(A)) =

m m dim(π(A)). Since we also have dim((Bs) ) = dim((Bs)Sym) = sm, and we know from Lemma

−1 m 1 that π(A) has codimension at least s, f (π(A)) must have codimension at least s in (Bs) .

m −1 −1 Let U = (Bs) \ f (π(A)). Since A and f (π(A)) have complex codimension at

m least s ≥ 2, by Proposition 12, both (Bs) \ A and U are simply-connected. Hence, f|U has

m a holomorphic lift fe|U : U → (Bs) \ A, where f|U = π ◦ fe|U . Since fe|U is bounded on U

−1 and f (π(A)) is an analytic set, by Riemann’s Continuation Theorem (Theorem 4), fe|U

m m extends to a holomorphic function fe :(Bs) → (Bs) with f = π ◦ fe.

m m (Bs) \ A (Bs) fe|U fe π π

m m m U (Bs)Sym \ π(A) (Bs) (Bs)Sym f|U f

m m It remains to show that fe :(Bs) → (Bs) is a proper map. However, since f = π ◦ fe is a proper map, it follows from Proposition 5 that fe is proper.

49 IV.5. End of the Proof of Theorem 3

m In Theorem 9, we characterized all proper holomorphic self-mappings of (Bs) , and

m m in Theorem 11, we showed that every proper holomorphic mapping (Bs) → (Bs)Sym lifts to

m a proper holomorphic self-mapping of (Bs) . We are now ready to combine these results to

m characterize proper holomorphic self-mappings of (Bs)Sym, which is the content of our main result, Theorem 3.

m m Proof. Let f :(Bs)Sym → (Bs)Sym be a proper holomorphic map. Then, since π is proper

m m and holomorphic, h = f ◦ π is a proper holomorphic map (Bs) → (Bs)Sym. By Theorem

m m 11, h lifts to a proper holomorphic map eh :(Bs) → (Bs) with h = π ◦ eh, and by Theorem

1 2 m σ(1) σ(2) σ(m) 9, we know eh has the structure eh(τ , τ , . . . , τ ) = (eh1(τ ), eh2(τ ),..., ehm(τ )) for

1 m all τ , . . . , τ ∈ Bs for some permutation σ of {1, 2, . . . , m}, where each ehi is a proper

m holomorphic self-mapping of (Bs) .

eh m m (Bs) (Bs)

π π h m m (Bs)Sym (Bs)Sym f

Since f ◦ π = π ◦ eh, and the left-hand side is invariant under the action of Sm on (τ 1, τ 2, . . . , τ m), we must have

σ(1) σ(2) σ(m) 1 2 m heh1(τ ), eh2(τ ),..., ehm(τ )i = heh1(τ ), eh2(τ ),..., ehm(τ )i (IV.12)

1 2 m m 2 m for every (τ , τ , . . . , τ ) ∈ (Bs) and every σ ∈ Sm. Let τ , . . . , τ ∈ Bs be fixed. Then, both sides of equation (IV.12) are functions only of τ 1, and since equation (IV.12) must hold

1 for all τ ∈ Bs and for all σ ∈ Sm, the ehi’s must all be identical. Thus, we conclude that h has the structure h(τ 1, τ 2, . . . , τ m) = hg(τ 1), g(τ 2), . . . , g(τ m)i, 50 where g = eh1 is a proper holomorphic self-mapping of Bs. Now, since f ◦ π = h, we have

f(hτ 1, τ 2, . . . , τ mi) = hg(τ 1), g(τ 2), . . . , g(τ m)i

m = gSym.

As noted in the introduction, for s ≥ 2, the only proper holomorphic self-mappings of Bs are the automorphisms of Bs. Hence, g is an automorphism of Bs and has the form given in (I.2).

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