Preface

"On ignore a peu pres tout de quels groupes peuvent etre groupes fondamentaux de

varietes algebriques,... " P. Deligne

(1987)1

This book was written because a lot can now be said about the fundamental

groups of algebraic varieties and of compact Kahler manifolds. Over the last few

years there has been a lot of progress on trying to understand which groups arise.

These developments were the topic of the Swiss Seminar (Borel Seminar) in the

Spring of 1995. To a large extent, this book is based on lectures given in the seminar,

although it is not, and is not meant to be, a faithful account of the seminar. We

try to explain what is currently known about the fundamental groups of compact

Kahler

manifolds2

and about some closely related questions. A lot of examples are

given to show that this class of groups is large and interesting. However, most of

the book is devoted to proving restrictions on these groups arising from the work

of Johnson-Rees, Gromov, Carlson-Toledo, Simpson and many others. In many

cases, especially when good accounts do not already exist, we give complete detailed

proofs; in other cases we prove special results and refer the reader to the literature

for the general case. The techniques used are a mixture of topology, differential

and algebraic geometry, and complex analysis.

Chapter 1, written mostly by D.K., is an introduction and overview, and it

explains the context of the problem to which this book is devoted. The section on

fundamental groups of compact complex surfaces contains a few new results which

have not appeared elsewhere.

Chapter 2, written mostly by D.K., discusses the general problem of finding a

holomorphic map inducing a given representation or homomorphism of the funda-

mental group of a compact Kahler manifold. In general, such a map does not exist,

but there are two notable exceptions. The first one is the quotient homomorphism

of the fundamental group to its first homology modulo torsion. The Albanese map

is a holomorphic map realising this. The second exception is representations onto

surface groups of genus at least two. We prove a theorem of Siu to the effect that

if the surface group is of maximal genus for the given manifold, then the represen-

tation is induced by a surjective holomorphic map with connected fibers from the

given Kahler manifold to a Riemann surface. More generally, the property of ad-

mitting some holomorphic map onto a closed hyperbolic Riemann surface is seen to

be a property of the fundamental group of a compact Kahler manifold. This allows

us to divide Kahler groups into so-called fibered and non-fibered groups. These

concepts tie in nicely with some of the techniques and results in later chapters. We

1[40],

Page 1

2Such

groups shall be called Kahler groups for short.