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Friday, September 22, 2017
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3:00pm [3:30pm]Prof. M. S. Raghunathan
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Title: Kodaira's theorem: criterion for embedding a compact Kahler manifold in projective space 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.

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