1. KAHLER GEOMETR Y 3
isotropic subspace to a maximal one, the genus of C may be larger than that of
C.
In this proof one sees another leitmotiv of this book, namely the consideration
of maps between a Kahler manifold X and an Eilenberg-Mac Lane space. This will
appear again and again.
We will prove the above Theorems 1.3, 1.4 and 1.5 in Chapter 2. The proof of
the Castelnuovo-de Franchis theorem is of course classical, but it will serve as an
introduction to the idea of constructing holomorphic maps from foliations defined
by holomorphic 1-forms. More sophisticated variants of that argument will appear
in Chapters 4 and 6. Siu's theorem gives rise to an important distinction between
two kinds of fundamental groups of compact Kahler manifolds, those which surject
onto surface groups, and those which do not.
We hope these examples have convinced the reader that Metatheorem 1.2 does
have substance. There are many other manifestations of it, for example the Al-
banese map, cf. Chapter 2, or the theory of variations of Hodge structure (Chap-
ter 7) and the Torelli theorems.
The attentive reader may have observed that the Theorem of Castelnuovo-
de Franchis was proved more than a quarter century before Kahler's paper, yet its
statement uses the concept of a Kahler metric! This discrepancy is due to the fact
that the Italians proved the result for complex algebraic surfaces, but the proof,
suitably phrased, works for any compact Kahler manifold.
In fact, smooth complex projective varieties are Kahler manifolds, because
CPn
is Kahler. This fact provides lots of examples, and it is possible that there
are, essentially, no others. See the Open Problem 1.7 in the next subsection.
In his paper, Kahler gives various examples of metrics satisfying du = 0. They
are all negatively curved in some sense; in particular he considers complex balls
and polydisks and their quotients, and he points out connections with the theory
of automorphic forms. But he does not use the fact that automorphic forms give
projective embeddings of the quotients, and he never bothers to look at CPn itself.
1.2. The modern point of view. From a modern point of view, the Kahler
form is defined without recourse to local coordinates. If J £ End(TX),
J2
= —Id,
is the almost complex structure on a manifold X with metric g, define
(3) u(X,Y) = g(JX,Y).
Considering triples (g, J, u) satisfying (3) and such that J is g- and u;-isometric,
any two entries determine the third one. Moreover, LU is always non-degenerate
because
u;n = nldvolg .
The Kahler condition du = 0 is an integrability condition which makes UJ into a
symplectic form. On the other hand, one wants the almost complex structure to
be integrable as well, namely one wants it to be induced from holomorphic charts.
By the Newlander-Nirenberg Theorem, this is equivalent to the vanishing of the
Nijenhuis tensor.
We can now define a Kahler manifold to be one that has compatible complex
and symplectic structures. A Riemannian metric is then given automatically by
reading (3) backwards. This is the Kahler metric.
Because the Kahler form of a Kahler metric is closed, it represents a cohomol-
ogy class [w] £
H2(X,
E). Moreover, because the top power of a; is a non-zero
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