# Lagrangian-invariant sheaves and functors for abelian varieties

### Alexander Polishchuk

University of Oregon, Eugene, USA

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## Abstract

We partially generalize the theory of semihomogeneous bundles on an abelian variety $A$ developed by Mukai, Semi-homogeneous vector bundles on an abelian variety, *J. Math. Kyoto Univ.* 18 (1978), 239–272. 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 $\mathcal{E}$ of $X_A\times X_A$ by $\mathbb{G}_m$. 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.