Lagrangian-invariant sheaves and functors for abelian varieties

2011/09/02 by Polishchuk, Alexander · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1109.0527

Abstract

We partially generalize the theory of semihomogeneous bundles on an abelian variety A developed by Mukai. This involves considering abelian subvarieties Y⊂ X_A=A×A and studying coherent sheaves on A invariant under the action of Y. The natural condition to impose on Y is that of being Lagrangian with respect to a certain skew-symmetric biextension of X_A× X_A. We prove that in this case any Y-invariant sheaf is a direct sum of several copies of a single coherent sheaf. We call such sheaves Lagrangian-invariant (or LI-sheaves). We also study LI-functors D^b(A)→ D^b(B) associated with kernels in D^b(A× B) that are invariant with respect to some Lagrangian subvariety in X_A× X_B. We calculate their composition and prove that in characteristic zero it can be decomposed into a direct sum of LI-functors. In the case B=A this leads to an interesting central extension of the group of symplectic automorphisms of X_A in the category of abelian varieties up to isogeny.

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• Derived isogenies between abelian varieties

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