Posts tagged maths | 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 where there are nice applications of complex analysis. In this post I will work over , 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 Read more...

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

Functor of points of non-affine schemes

Posted by Martin Orr on Saturday, 07 November 2009 at 16:59

This post was inspired by Monday's algebraic geometry exercise class, although in fact it fits neatly into my series on functors of points (except that it requires you to know what a scheme is, while previously I have considered only affine schemes). I shall prove the following theorem:

Theorem. There is a canonical bijection between morphisms of -schemes and natural transformations of the corresponding functors of points.

3 comments Tags alg-geom, maths, points-func Read more...

Functors of points and base ring

Posted by Martin Orr on Thursday, 08 October 2009 at 09:58

So far in my series on functors of points, I have considered functors for some fixed field . We begin by observing that we may allow to be any ring. Then I consider whether it is possible to relate functors with base ring to functors with base ring , with only partial success.

no comments Tags alg-geom, maths, points-func Read more...

Morphisms and functors of points

Posted by Martin Orr on Thursday, 01 October 2009 at 15:45

This post will discuss the fact that -points of an affine -scheme (and more general objects) are the same as morphisms . James already brought this up in his comment last time. As well as proving this in the affine -scheme case, I shall attempt to give an intuitive explanation of this fact, although I don't find this entirely satisfying.