Lagrangian-invariant sheaves and functors for abelian varieties

TL;DR

This paper extends the theory of semihomogeneous sheaves on abelian varieties by introducing Lagrangian-invariant sheaves and functors, revealing their structure and composition properties, especially over characteristic zero fields.

Contribution

It introduces the concept of LI-sheaves and LI-functors, generalizing semihomogeneous bundles and analyzing their composition and symmetries in the context of abelian varieties.

Findings

LI-sheaves decompose into copies of a single sheaf

LI-functors decompose into sums of LI-functors in characteristic zero

Identifies a central extension of symplectic automorphisms group

Abstract

We partially generalize the theory of semihomogeneous bundles on an abelian variety $A$ developed by Mukai. This involves considering abelian subvarieties $Y\subset X_A=A\times\hat{A}$ 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 $X_A\times X_A$. 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 $D^b(A)\to D^b(B)$ associated with kernels in $D^b(A\times B)$ that are invariant with respect to some Lagrangian subvariety in $X_A\times X_B$. 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.