x PREFACE

give complete proofs based on classical arguments (Albanese map, Castelnuovo-de

Franchis Theorem), which are toy versions of more delicate arguments in the same

style which are used in later chapters.

Chapter 3, written mostly by J.A., applies the techniques of real homotopy

theory to study Kahler groups. Rather than looking at the fundamental group

itself, we look here at its real Malcev completion. This approach goes back to the

work of Sullivan and of Deligne-Griffiths-Morgan-Sullivan in the 1970s, but there

are a number of new results as well.

Chapter 4, written mostly by M.B., applies

L2-cohomology

to prove restrictions

on the fundamental groups of Kahler manifolds, following an idea of Gromov and the

elaborations on it by Arapura-Bressler-Ramachandran. We give a careful account

of Gromov's theorem showing that Kahler groups cannot split as free products.

More generally, we show that Kahler groups have finitely many ends. These results

are proved by constructing holomorphic fibrations over curves, generalising the

results of Chapter 2.

Chapter 5 gives an outline of some existence theorems for harmonic maps.

These are needed for the applications in Chapters 6 and 7. First we outline a proof

of the theorem of Eells-Sampson, giving existence of harmonic maps in homotopy

classes of maps whose target has a non-positively curved Riemannian metric. Then

we explain the generalisation of this result to twisted or equivariant harmonic maps

due to Corlette, Donaldson and Labourie.

Chapter 6, written mostly by D.T., applies harmonic maps to the study of

Kahler groups. It begins with a proof of the Siu-Sampson Bochner formula, which

implies that certain harmonic maps are in fact pluriharmonic. Combining this with

the existence theorems of Chapter 5, we have a large supply of pluriharmonic maps

from Kahler manifolds to negatively curved manifolds. Following Carlson-Toledo,

Siu and Sampson, we prove a general factorisation theorem for such maps, which

has a number of geometric corollaries. These include a proof of Siu's theorem that

was proved using more classical methods in Chapter 2, and many restrictions on

Kahler groups. For example, it is shown that fundamental groups of real hyperbolic

manifolds of dimension at least three cannot be fundamental groups of compact

Kahler manifolds. The final section of this chapter discusses geometric applications

of more general harmonic maps, maps for which the target space is a negatively

curved space which need not be a manifold. This more general existence theorem

is not covered in Chapter 5, and we refer to the original paper by Gromov-Schoen

for it.

Chapter 7, written mostly by K.C., is an introduction to the non-Abelian

Hodge theory of Corlette and Simpson. This uses the existence theorems for har-

monic maps in Chapter 5. There is a detailed discussion of the Riemann surface

case due to Hitchin. This motivates the general case, the details of which are often

omitted and replaced by references to the original papers. Some applications to

fundamental groups are given following Simpson. At the end we present Reznikov's

recent proof of the Bloch conjecture.

Chapter 8, written mostly by D.T., gives a number of very non-obvious ex-

amples of groups which occur as Kahler groups, in fact, as fundamental groups of

smooth complex projective varieties. This includes non-Abelian nilpotent groups,

and some groups which are not residually finite. Some of the examples in this

chapter have not appeared elsewhere.