Martin's Blog

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

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

Mumford-Tate groups

Posted by Martin Orr on Monday, 04 October 2010 at 12:37

In this post I will define the Mumford-Tate group of an abelian variety. This is a \mathbb{Q}-algebraic group, such that the Hodge structure is a representation of this group. The Mumford-Tate group is important in the study of Hodge theory, and surprisingly also tells us things about the \ell-adic representations attached to the abelian variety.

4 comments Tags abelian-varieties, alg-geom, alg-groups, hodge, maths

Hodge structures and abelian varieties

Posted by Martin Orr on Friday, 24 September 2010 at 08:48

I spend most of my time thinking about the Hodge structures attached to abelian varieties, so I decided that I should explain what these Hodge structures are. A Hodge structure is a type of algebraic structure found on the (co)homology of complex projective varieties.

Here I will discuss only the special case of the first homology of abelian varieties. This is the simplest case, but is nonetheless very important. In particular, the Hodge structures on other homology and cohomology groups for abelian varieties can be calculated from that of the first homology. Also Hodge structures on the first (but not higher) cohomology of non-abelian varieties can be reduced to the case of abelian varieties by passing to something called the Albanese variety, generalising the Jacobian of curves.

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

Hodge theory talk

Posted by Martin Orr on Wednesday, 16 December 2009 at 19:48

Last week I gave a talk on Hodge theory. For the Differential Geometry course, all the students have to give a talk on a topic related to the course. The talk was very long - 1 hour 45 minutes - but this is about the average length of the talks so far. I did my best to shorten it by leaving out unimportant details. Had it not been for the fact that many other talks were longer, I would have removed sections of it entirely, but it did cover about the minimum needed to reach a point of interest to me as an algebraic geometer.

This was the first time I have given a talk of any length in French. This was not too difficult, as I had practised the talk, but probably did slow me down a bit. I am sure the language was far from perfect; for example, I probably should have used the subjunctive all over the place but I didn't bother with it. But the audience were not too concerned about that.

The first half of the talk contained a lot of analysis, needed to prove the Hodge theorem. This is not my area, but it was fun to learn a little bit; I skipped out all the tedious calculations. The second half contained applications of this to complex manifolds, leading up to the fundamental example of a Hodge structure. I shall need soon to learn about the latter in a more abstract setting; no doubt preparing this talk has given me some of the motivation for them, but I am not sure how useful all the proofs will turn out to be.

no comments Tags hodge, languages, m2, maths, talk

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