Description
Title: Kodaira's theorem: criterion for embedding a compact Kahler
manifold in projective space (Lecture 2)
Abstract: Let $M$ be a compact Kahler manifold and $\Omega (M)$ the
canonical $2$-form on $M$. When $M$ is projective $n$-spce $\P^n(\C)$ ,
$H^2(M,\C)$
is of dimension 1. It follows that for any Kahler metric on the projective
space, the cohomology class $[\Omega (M)$ of the canonical $2$-form is a
multiple of the (unique up to sign) of a generator of $H^2(M,\Z)$. It is
immediate from this that if $M$ is a complex sub-manifold of $\P^n(\C)$ for
some $n$, then for the Kahler metric on $M$ induced from one on $\P^n(\C)$,
it is clear that $[\Omega(M)] \in $\C \cdot H^2(M, Z)$. Kodaira's theorem
is a converse to this fact: If a complex manifold $M$ admits a Kahler
metric such that the class of $\Omega(M)$ is a multiple of an integral
class, then $M$ can be embedded in some projective space. This result was
conjectured by W V D Hodge.