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.

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.

Description

Room 215, Department of Mathematics

Date

Fri, October 6, 2017

Start Time

3:30pm IST

Priority

5-Medium

Access

Public

Created by

DEFAULT ADMINISTRATOR

Updated

Fri, October 6, 2017 11:56am IST