Martin Orr's Blog

Complex abelian varieties and Riemann forms

Posted by Martin Orr on Wednesday, 30 December 2009 at 21:48

The theory of abelian varieties is very beautiful, both in its arithmetic and geometrical aspects, and also looking just over \mathbb{C} where there are nice applications of complex analysis. In this post I will work over \mathbb{C}, and sketch a proof that a complex torus is isomorphic to an abelian variety if and only if it admits a Riemann form. This will assume some knowledge of the theory of complex manifolds.

1 comment Tags abelian-varieties, alg-geom, 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