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.
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