Martin's Blog

Line bundles and morphisms to the dual variety

Posted by Martin Orr on Saturday, 28 May 2011 at 15:25

Over the complex numbers, the dual of an abelian variety A is defined to have a Hodge structure dual to that of A. Hence morphisms A \to A^\vee can be interpreted as bilinear forms on the Hodge structure of A. Of particular importance are the morphisms corresponding to Hodge symplectic forms.

Last time we saw that A^\vee can also be interpreted as a group of line bundles on A. Today we will use this interpretation to define morphisms \phi_\mathcal{L} : A \to A^\vee which turn out to be the same as those corresponding to Hodge symplectic forms. Then we generalise the definition of \phi_\mathcal{L} to base fields other than \mathbb{C}, which we will use next time in constructing dual abelian varieties over number fields.

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

Dual abelian varieties and line bundles

Posted by Martin Orr on Monday, 09 May 2011 at 14:30

The definition I gave last time of dual abelian varieties was very much dependent on complex analytic methods. In this post I will explain how dual varieties can be interpreted geometrically: the points of A^\vee correspond to a certain group of line bundles on A. We construct a single line bundle \mathcal{P} on the product A \times A^\vee, the Poincaré bundle, such that all line bundles on A arise as restrictions of \mathcal{P}, and show that the pair (A^\vee, \mathcal{P}) satisfies a universal property.

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

Dual abelian varieties over the complex numbers

Posted by Martin Orr on Tuesday, 26 April 2011 at 12:35

In this post I will define dual abelian varieties over the complex numbers. The motivation is that polarisations can be interpreted as isogenies from an abelian variety to its dual. For the moment, all this is tied to Hodge structures so only works over the complex numbers, but this is the view of polarisations which will we will generalise later to other fields.

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

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...

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