Martin's Blog

The Hodge filtration and universal vector extensions

Posted by Martin Orr on Friday, 13 June 2014 at 20:10

We will begin this post by looking at the isomorphism between the Hodge filtration  H^{0,-1}(A) \subset H_1(A, \mathbb{C}) of a complex abelian variety A and the natural filtration  T_0(A^\vee)^\vee \subset T_0(E_A) on the tangent space to the universal vector extension of A.

The significance of this isomorphism is that the Hodge filtration, as we defined it before, is constructed by transcendental methods, valid only over \mathbb{C}, but the universal vector extension is an object of algebraic geometry. So this gives us an analogue for the Hodge filtration for abelian varieties over any base field. Furthermore, in the usual way of algebraic geometry, the construction of the universal vector extension can be carried out uniformly in families of abelian varieties.

We will use the construction of the universal vector extension in families to show that “the Hodge filtration varies algebraically in families.” We will first have to explain what this statement means. We will also mention briefly why H^{-1,0}(A) does not vary algebraically.

A note on the general philosophy of this post: the usual construction of an algebraic-geometric object isomorphic to the Hodge filtration uses de Rham cohomology, which works for H^n of an arbitrary smooth projective variety. My aim in using universal vector extensions is to give an ad hoc construction of de Rham (co)homology, valid only for H_1 of an abelian variety, requiring less sophisticated technology than the general construction. This fits with previous discussion on this blog of the Hodge structure on H_1, constructed via the exponential map from the tangent space of A, and of the \ell-adic H_1, constructed as the Tate module.

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Rendering OpenStreetMap with Tilemill

Posted by Martin Orr on Tuesday, 20 May 2014 at 19:30

I have decided to play around with creating some maps using the freely available data from OpenStreetMap. I am mainly interested in creating maps of London bus routes but first I had to learn how to use the Tilemill rendering software and how to load OpenStreetMap data into Tilemill. In this post I will describe how I rendered my first map, a very simple line map of London.

My first map: Line map of London

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Universal vector extensions of abelian varieties

Posted by Martin Orr on Friday, 09 May 2014 at 18:25

In the last post we showed that extensions of an abelian variety A by the additive group \mathbb{G}_a are classified by a vector space \operatorname{Ext}^1(A, \mathbb{G}_a) which is canonically isomorphic to T_0(A^\vee). In this post I will show that there is a so-called universal vector extension of A, that is, a vector extension E_0 of A such that every vector extension of A can be obtained in a unique way as a pushout of E_0. The vector group part of E_0 is \operatorname{Ext}^1(A, \mathbb{G}_a)^\vee, and an ingredient which we require from last time is that this is finite-dimensional.

Even if you are not interested in vector extensions of abelian varieties for their own sake (and I am not), they still have interesting properties. For example, I will show that they provide examples of algebraic varieties over the complex numbers which are non-isomorphic as algebraic varieties but which are isomorphic as complex manifolds.

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Vector extensions of abelian varieties

Posted by Martin Orr on Thursday, 17 April 2014 at 18:12

A theme of my posts on abelian varieties has been ad hoc constructions of objects which are equivalent to the (co)homology of abelian varieties together with their appropriate extra structures -- the period lattice for singular homology and the Hodge structure, the Tate module for \ell-adic cohomology and its Galois representation. I want to do the same thing for de Rham cohomology. To prepare for this, I need to discuss vector extensions of abelian varieties -- that is extensions of abelian varieties by vector groups.

In this post I will define and classify extensions of an abelian variety by the additive group. We will conclude that \operatorname{Ext}^1(A, \mathbb{G}_a), the set of isomorphism classes of such extensions, forms a vector space isomorphic to the tangent space of the dual of A. Most of this was discovered by Rosenlicht in the 1950s.

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Endomorphisms of simple abelian varieties

Posted by Martin Orr on Thursday, 03 January 2013 at 15:51

Today I will discuss the classification of endomorphism algebras of simple abelian varieties. The endomorphism algebra of a non-simple abelian variety can easily be computed from the endomorphism algebras of its simple factors. For a simple abelian variety, its endomorphism algebra is a division algebra of finite dimension over \mathbb{Q}. (A division algebra is a not-necessarily-commutative algebra in which every non-zero element is invertible.) As discussed last time, the endomorphism algebra also has a positive involution, the Rosati involution. There may be many Rosati involutions, coming from different polarisations of the abelian variety, but all we care about today is the existence of a positive involution. Division algebras with positive involutions were classified by Albert in the 1930s.

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