Lagrangian-invariant sheaves and functors for abelian varieties

  • Alexander Polishchuk

    University of Oregon, Eugene, USA
Lagrangian-invariant sheaves and functors for abelian varieties cover
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We partially generalize the theory of semihomogeneous bundles on an abelian variety AA developed by Mukai, Semi-homogeneous vector bundles on an abelian variety, J. Math. Kyoto Univ. 18 (1978), 239–272. This involves considering abelian subvarieties YXA=A×A^Y\subset X_A=A\times\hat{A} and studying coherent sheaves on AA invariant under the action of YY. The natural condition to impose on YY is that of being Lagrangian with respect to a certain skew-symmetric biextension E\mathcal{E} of XA×XAX_A\times X_A by Gm\mathbb{G}_m. We prove that in this case any YY-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 Db(A)Db(B)D^b(A)\to D^b(B) associated with kernels in Db(A×B)D^b(A\times B) that are invariant with respect to some Lagrangian subvariety in XA×XBX_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=AB=A this leads to an interesting central extension of the group of symplectic automorphisms of XAX_A in the category of abelian varieties up to isogeny.