Page 164 - Curvature and Homology

CHAPTER V

In a well-known manner one can associate with an irreducible curve V,
a real analytic manifold M2 of two dimensions called the Riemann
surface of Vl. Since the geometry of a Riemann surface is conformal
geometry, M2 is not a Riemannian manifold. However, it is possible to
define a Riemannian metric on MZ in such a way that the harmonic
forms constructed with this metric serve to establish topological in-
variants of Me. In his book on harmonic integrals 1391, Hodge does
precisely this, and in fact, in seeking to associate with any irreducible
algebraic variety V, a Riemannian manifold MZn of 2n dimensions he is
able to obtain the sought after generalization of a Riemann surface
referred to in the introduction to Chapter I. The metric of an Mw has
certain special properties that play an important part in the sequel
insofar as the harmonic forms constructed with it lead to topological
invariants of the manifold. The approach we take is more general and
in keeping with the modern developments due principally to A. Weil
[70, 721. We introduce first the concept of a complex structure on a
separable Hausdorff space M in analogy with 5 1 .l. In terms of a
given compl&x structure a Riemannian metric may be defined on M.
If this metric is torsion free, that is, if a certain 2-form adsociated with
the metric and complex structure is closed, the manifold is called a
Kaehler manifold. As examples, we have complex projective n-space P,
and, in fact, any projective variety, that is irreducible algebraic variety
holomorphically imbedded without singularities in P,.
The local geometry of a Kaehler manifold is studied, and in Chapter VI,
from these properties, its global structure is determined to some extent.
In Chapter VII we further the discussions of Chapter I11 by considering
groups of transformations of Kaehler manifolds-in particular, Kaehler-
Einstein manifolds. It may be said that of the diverse applications of the
theory of harmonic integrals, those made to Kaehler manifolds are
amongst the most interesting.