Martin's Blog

Polarisable complex tori are projective

Posted by Martin Orr on Saturday, 26 March 2011 at 16:07

Last time, we defined polarisations on H_1 Hodge structures and saw that if A is a complex abelian variety, then H_1(A) has a polarisation. This time we will prove the converse: if X is a complex torus such that H_1(X) has a polarisation, then X is an abelian variety (in other words, X can be embedded in projective space). The proof is based on studying invertible sheaves on X.

This is long, even though I have left out all the messy calculations. For full details, see Mumford's Abelian Varieties or Birkenhake-Lange's Complex Abelian Varieties. For the next post, you will only need to know the two statements labelled as theorems.

This theorem is a special case of the Kodaira Embedding Theorem, which tells you that any compact complex manifold is projective if it has a polarisation, but that is somewhat more difficult.

6 comments Tags abelian-varieties, alg-geom, hodge, maths