Martin's Blog

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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