As a first application of this argument, one sees that equation (2) implies equa-
tion (1) which is trivially true in the flat metric. Conversely, a calculation shows
that, given (1), one can find new coordinates in which the metric takes the form
Scholium 1.1 has a provocative consequence, which is one of the leitmotives of
this book:
1.2. Kahler manifolds are complex manifolds whose geometry
reduces to linear algebra.
This sounds ridiculous, but it has much more substance than one can possibly
imagine from a naive point of view. In fact, not only the local geometry but
the global analytic geometry and the algebraic topology of a Kahler manifold are
controlled by linear algebra. We will see in Chapter 3 that a lot of potentially
quite subtle information about the homotopy type of a Kahler manifold is actually
determined by, and can be computed from, the de Rham algebra of differential
forms on the manifold (or its cohomology). This algebra is clearly an object of
linear algebra, and it is extremely close in spirit to Kahler's original idea of deriving
invariants using differential forms.
Here is another example of the reduction to linear algebra, again through dif-
ferential forms:
1.3 (Castelnuovo-de Franchis (1905)). Let X be a compact Kahler
manifold, and let u)\,... , uor be linearly independent holomorphic 1 -forms with
uji A ujj; = 0 for all i,j. Then there exists a holomorphic map f:X-+C, where C
is a complex curve, such that uj\,... , uur are in the image of the pullback /*.
Prom this one can deduce the following statement, see [29], which is even closer
to Metatheorem 1.2:
1.4 (Catanese (1989)). Let X be a compact Kahler manifold. There
exists a compact complex curve C of genus g 2 and a surjective holomorphic map
f: X C if and only if there exists a g-dimensional maximal isotropic subspace
where isotropic means that the image of
is zero.
Perhaps surprisingly, these are really statements about the fundamental group
of X, as was proved first by Siu [122] and later, independently, by Beauville [10].
1.5 (Siu (1987), Beauville (1988)). LetX be a compact Kahler man-
ifold and g 2 an integer. Then X admits a non-constant holomorphic map to
some compact Riemann surface of genus g' g having connected fibers if and only
if there is a surjective homomorphism h:
7Ti(Cg), with 7Ti(Cg) the funda-
mental group of a compact Riemann surface of genus g.
PROOF. If the holomorphic map exists, it induces a surjection on 7T], and the
surface group of genus g' g surjects onto the surface group of genus g.
Conversely, suppose we are given h. Then because C is an Eilenberg-Mac Lane
space K(ni(C), 1), there exists a homotopy class of continuous maps / : X C
inducing h. But the algebra homomorphism /*: i/*(C,C) H*(X,C) is injective
in degree 1 and maps isotropic subspaces of Hl(C,C) to isotropic subspaces of
Hl(X,C). Every isotropic subspace is contained in a maximal one. Given the
maximal isotropic subspace, Theorem 1.4 shows that there exists a non-constant
holomorphic map from A^ to a curve C". Because one passes from an arbitrary
Previous Page Next Page