Martin Orr'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 Read more...

Polarisations on Hodge structures

Posted by Martin Orr on Saturday, 26 February 2011 at 18:27

In the last post, I discussed Hodge symplectic forms. Now I shall show that the H_1 of an abelian variety has a polarisation, which is defined to be a Hodge symplectic form satisfying a positivity condition. The importance of polarisations is that they give a way of recognising which H_1 Hodge structures come from abelian varieties - I shall discuss this application next time.

2 comments Tags abelian-varieties, alg-geom, hodge, maths Read more...

Hodge symplectic forms

Posted by Martin Orr on Saturday, 18 December 2010 at 15:00

Both the Hodge structure and the Tate module of an abelian variety come with symplectic forms which are (almost) preserved by the action of the relevant group (Mumford-Tate or Galois group). The form on the Tate module, called the Weil pairing, will require some preparation. So in this post I will construct the Hodge symplectic forms (also called the Riemann forms) on the Hodge structure. Next time I will discuss some further properties of Hodge forms.

no comments Tags abelian-varieties, alg-geom, hodge, maths Read more...

Images of Galois representations

Posted by Martin Orr on Saturday, 27 November 2010 at 16:22

In this post, I will continue to talk about the \ell-adic representations attached to abelian varieties, and in particular the images G_\ell of these representations. I will define algebraic groups approximating G_\ell, which are often more convenient to work with. I will end by stating the Mumford-Tate conjecture, linking G_\ell to the Mumford-Tate group.

7 comments Tags abelian-varieties, alg-geom, alg-groups, hodge, maths Read more...

Tate modules

Posted by Martin Orr on Sunday, 21 November 2010 at 17:32

I said after my last post that I would write something about \ell-adic representations coming from abelian varieties. I have finally got around to doing so: here I will tell the story of how these representations are defined, and show that the Tate module is canonically isomorphic to H_1(A, \mathbb{Z}) \otimes \mathbb{Z}_\ell. Next time I will relate this to Mumford-Tate groups.

2 comments Tags abelian-varieties, alg-geom, maths, number-theory Read more...

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