# Shafarevich Maps and Automorphic Forms

JÁNOS KOLLÁR
Series: Porter Lectures
Pages: 212
https://www.jstor.org/stable/j.ctt7ztqb4

1. Front Matter
(pp. i-iv)
(pp. v-vi)
3. PREFACE
(pp. vii-viii)
4. ACKNOWLEDGMENTS
(pp. ix-2)
5. INTRODUCTION
(pp. 3-16)

The aim of these notes is to study algebraic varieties whose fundamental group is nontrivial.

Fundamental groups of algebraic varieties have been studied quite extensively. Most of the research has centered on trying to determine which groups occur as fundamental groups of algebraic varieties. This problem is very interesting, and there have been numerous important results. [Arapura94] is a recent overview of this direction.

My main interest is to see how the presence of a “large” fundamental group influences other algebro-geometric properties of a variety. The main examples I have in mind are the Kodaira dimension and the pluricanonical maps....

6. ### PART I. SHAFAREVICH MAPS

• CHAPTER 1 Lefschetz-Type Theorems for π1
(pp. 19-26)

LetXbe a smooth projective variety andHXa smooth hyperplane section (under some projective embeddingX⊂ ℙ). Lefschetz theory asserts that the topology ofXand the topology ofHare closely related. One of the simplest results is the following.

1.1 Theorem. [Lefschetz24]Let X be a smooth projective variety and HX a smooth hyperplane section. Thenπ1(H) → π1(X)is an isomorphism ifdimH≥ 2and a surjection ifdimH= 1.

We try to extend this result to arbitrary subvarietiesZXby connecting it to...

• CHAPTER 2 Families of Algebraic Cycles
(pp. 27-35)

In this chapter I will set up the machinery of families of algebraic cycles on a varietyX. Since we deal with properties ofXat very general points only, we are able to ignore all the subtle aspects of cycle theory. Almost everything that we need about cycles is contained in [Hodge-Pedoe52].

2.1 Definition. LetXbe a normal variety. By anormal cycleonXwe mean an irreducible and normal varietyWtogether with a finite morphismw : W → X, which is birational to its image.

LetZXbe any closed irreducible subvariety,...

• CHAPTER 3 Shafarevich Maps and Variants
(pp. 36-48)

In this chapter I will prove (1.6), (1.8), and some other versions as well.

3.1 Definition. LetGbe a group andH1,H2subgroups. We say thatH1isessentially a subgroupofH2ifH1H2has finite index inH1. We denote this relationship byH1H2.

By definition,H≲ {1} iffHis finite andGHiffHhas finite index inG.

We need the following easy result:

3.2 Proposition.Let X be a normal variety. Consider the following diagram:$\begin{matrix} \ \ \ \ \ \ \ \ \ \ T \textamp \overset{i}\longrightarrow \textamp \ \ V \textamp \overset{w}\longrightarrow \textamp Z \textamp \overset{n}\longrightarrow \textamp X \\ \downset{p}\downarrow \textamp \textamp \\ \ \ V \end{matrix}$where all the varieties are irreducible and normal, pi and...

• CHAPTER 4 The Fundamental Group and the Classification of Algebraic Varieties
(pp. 49-56)

In this chapter I will outline the relationship between the fundamental group of an algebraic variety and other properties and invariants.

If we take the product of a varietyX1with a simply connected varietyX2, then the fundamental group ofX1×X2does not carry any information aboutX2. Before we can say anything substantial, we must decide when to expect the fundamental group to govern global properties of the whole variety.

The general outlines of this problem are clear, but it is difficult to pin down the best version at the moment. We use several variants. There...

7. ### PART II. AUTOMORPHIC FORMS:: CLASSICAL THEORY

• CHAPTER 5 The Method of Poincaré
(pp. 59-70)

LetCbe a Riemann surface andp:M → Cits universal covering space.Mis a simply connected Riemann surface, thus it is either ℂℙ1, ℂ or the unit disc ∆. In all three cases we understand the function theory ofMquite well. The aim of the theory of automorphic forms is to connect the function theory ofMand the function theory ofC. There are several ways to do this and we explore some of the possibilities.

5.1 Naive approach. LetCbe a curve of genus at least 2 with universal cover ∆....

• CHAPTER 6 The Method of Atiyah
(pp. 71-80)

LetXbe a compact Kähler manifold andEa vector bundle onXwith a Hermitian metric. By Hodge theory, the cohomologyHι(X,E) can be represented byE-valued harmonic (0,i)-forms. Let$\tilde{X}$be the universal cover ofX. We can pull back the Kähler form,Eand its metric to$\tilde{X}$. The definitions of Hodge theory being local, harmonic forms make sense on$\tilde{X}$. Thus we conclude thatHι(X,E) can be represented by-valued harmonic (0,i)-forms on$\tilde{X}$which are invariant under π1(X).

Unfortunately π1(X)-invariant forms are not in anyLpspace (p<...

• CHAPTER 7 Surjectivity of the Poincaré Map
(pp. 81-91)

We keep the notation of chapter 5. Given a complex manifordXand a vector bundleE, we take$\tilde{X}$,as before. Let$P : (L^{1} \cap H)(\tilde{X}, \tilde{E}) \rightarrow H^{0} (X, E)$be the Poincaré map (5.10). Our aim is to try to see ifPis surjective.

Before we consider the surjectivity of the Poincaré series, we define Bergman kernels in a general setting.

(7.1.1) LetMbe a complex manifold andEa vector bundle onM. We want to define an inner product on sections ofE.

The simple way is to consider a volume formdmonMand Hermitian metrich(...

• CHAPTER 8 Ball Quotients
(pp. 92-102)

In order to illustrate some of the previous methods, we compute in detail the example of the unit ball in ℂn. This is the easiest example of a Hermitian homogeneous space.

8.1 Definition. Consider the standard action ofSL(n+ 1, ℂ) on ℂℙn. We write coordinates on ℂℙnas (x0: … :xn). On the affine chartx0≠ 0 we introduce affine coordinateszι=xι/x0(i≠ 0). LetSU(1,n) <SL(n+ 1, ℂ) be the subgroup that leaves the Hermitian form$Q = - x_{0}\bar{x}_{0} + x_{1}\bar{x}_{1} + \cdots + x_{n}\bar{x}_{n}$invariant. In the above affine chartQ= 0 can be written...

8. ### PART III. VANISHING THEOREMS

• CHAPTER 9 The Kodaira Vanishing Theorem
(pp. 105-114)

In this chapter I will present a simple proof of the Kodaira vanishing theorem and some of its generalizations. More subtle vanishing theorems will be treated in subsequent chapters.

(9.1.1) Let M be a compact complex manifold and L a line bundle on M with a Hermitian metric h whose curvature form is positive definite everywhere. Then

Hι(M,KML) = 0for i> 0.

(9.1.2) Let X be a smooth projective variety and L an ample line bundle on X. Then

Hι(X,KXL) = 0for i> 0.

The two versions are equivalent, but this misses...

• CHAPTER 10 Generalizations of the Kodaira Vanishing Theorem
(pp. 115-126)

In the last decade it became increasingly clear that it is frequently very useful to have analogs of (9.1) and (9.14) for nonample line bundles. The general philosophy suggests that similar results should hold if the line bundle is a “small perturbation” of an ample line bundle. Two notions of perturbation emerged, one in algebraic and one in complex differential geometry. First we recall these definitions and then we prove that they are essentially equivalent. We restrict our attention to smooth varieties, though both versions can be treated on singular spaces as well.

10.1 Definition.LetXbe a smooth...

• CHAPTER 11 Vanishing of L²-Cohomologies
(pp. 127-132)

In this chapter I will give a quick review of the results of [Demailly82,92] about vanishing theorems for line bundles with singular Hermitian metrics.

11.1 Definition-Proposition. (11.1.1) LetXbe a complex manifold. By adegenerate Kähler formwe mean a closed positive Kähler form ω defined on a Zariski dense open setVXsuch that if (z1, … ,zn) is any local coordinate chart onX, then$\omega = \sum g_{ij}dz_{i} \wedge d \bar{z}_{j}$and thegijare bounded. We denote bydV(ω) =in(n–1)ωnthe correspondingdegenerate volume form. (The definition is dictated by immediate needs rather than by any...

• CHAPTER 12 Rational Singularities and Hodge Theory
(pp. 133-138)

This chapter is rather technical and its results are not used elsewhere in these notes.

In the previous discussions about vanishing theorems, the starting point was the surjectivity of the natural mapHι(X, ℂ) →Hι(X, OX) for smooth projective varieties over ℂ. As in (10.11) it is frequently desirable to have vanishing theorems on singular spaces as well. By (9.12) the surjectivity ofHι(X, ℂ) →Hι(X, OX) for a class of singular varieties implies some vanishing theorems. We consider the case whenXhas rational singularities.

LetXbe an arbitrary proper variety over ℂ. Let ℂXdenote...

9. ### PART IV. AUTOMORPHIC FORMS REVISITED

• CHAPTER 13 The Method of Gromov
(pp. 141-150)

One of the drawbacks of (5.22) is that it assumes the existence of bounded holomorphic functions onM. In several interesting cases this is not satisfied or not known. One such example is whenXis an Abelian variety andM= ℂn. In this chapter I will present the method of [Gromov91], which does not assume anything about the existence of holomorphic functions onM. The method is fairly general and can best be formulated for group actions on topological spaces.

13.1 Notation. LetMbe a topological space andLa complex line bundle onM. Let Γ...

• CHAPTER 14 Nonvanishing Theorems
(pp. 151-160)

In chapters 9–11 we proved several theorems which assert that under suitable conditions the higher cohomology groups of line bundles are zero. If a varietyXand a line bundleLsatisfy these assumptions, then

(14.1)h0(X,KXL) = χ(X,KXL).

A result of this type can be used in two ways.

(14.2.1) As a corollary we obtain that χ(X,KXL) ≥ 0, which translates to an inequality between Chern classes ofXandLby the Hirzebruch-Riemann-Roch formula (6.1.1). An application along this line is given in chapter 17.

(14.2.2) In some...

• CHAPTER 15 Plurigenera in Etale Covers
(pp. 161-166)

LetXbe a smooth projective variety andNa line bundle onX. The methods of vanishing and nonvanishing apply toNif we can writeNK+M+ ∆, whereMis nef and big and (X, ∆) is klt. Thus arises the question: can one write every line bundle in this form? IfNis very negative then this is impossible, so we are primarily interested in line bundles of the formKMwhereMis big. Even for such divisors there are easy counter examples.

The problem becomes more interesting if...

• CHAPTER 16 Existence of Automorphic Forms
(pp. 167-172)

LetXbe a smooth projective variety and π :MXits universal covering space. The classical theory of automorphic forms deals with some cases whereMis well known, for instance a bounded domain in ℂn. One can thus use the knowledge ofMto construct sections of line bundles onX. The most frequently studied case is for powers of the canonical bundle. The following more general situation is a natural extension of this.

16.1 Definition.LetXbe a (smooth) projective variety and π :MXa (usually infinite) covering space corresponding to...

10. ### PART V. OTHER APPLICATIONS AND SPECULATIONS

• CHAPTER 17 Applications to Abelian Varieties
(pp. 175-182)

The simplest examples of varieties with generically large algebraic fundamental group are those that admit a generically finite morphism to an Abelian variety. In this chapter we will study such varieties. This illustrates the use of the general methods, frequently in a simpler version.

The first result is a further improvement of [Kollár93b, 1.17]. Its predecessors include [Ueno75; Kawamata-Viehweg80; Kawamata81; Kollár86a]. (b1(X) = dimH1(X, ℚ) is the first Betti number.)

17.1 Theorem.Let X be a smooth proper variety overℂ.The following are equivalent:

(17.1.1) X is birational to an Abelian variety;

(17.1.2) b1(X) = 2 dimX...

• CHAPTER 18 Open Problems and Further Remarks
(pp. 183-190)

18.1 The Shafarevich conjecture.I still have not done anything about finding holomorphic functions on universal covers. The main aim of these results is rather to go around the Shafarevich conjecture.

One could say that the Shafarevich conjecture has two parts. The first part is a philosophical statement that varieties with “large” fundamental group form a rather special and important subclass of all varieties. The present notes give ample support to this assertion. The second part is a precise statement that “large” fundamental group is equivalent to the universal cover being Stein. As far as I can tell, my methods...

11. REFERENCES
(pp. 191-200)
12. INDEX
(pp. 201-201)