Symplectic Biextensions and a Generalization of the Fourier-mukai Transform

Let A be an abelian variety,  be the dual abelian variety. The Fourier-Mukai transform is an equivalence between the derived categories of coherent sheaves Db(A) and Db(Â). Notice that there is a ”symplectic” line bundle LA on (Â×A)2, namely, LA = p14P⊗√∈∋P where P is the Poincaré line bundle on Â×A, such that the standard embeddings A ⊂ Â×A and  ⊂ Â×A are ”lagrangian” with respect to LA, i.e. the restrictions of LA to A 2 and  are trivial (and they are maximal with this property). The purpose of this paper is to establish an analogous equivalence of derived categories for arbitrary lagrangian subvarieties in an abelian variety X equipped with a line bundle L over X which satisfies some properties similar to that of LA (L should be a symplectic biextension—see below). Namely, with every lagrangian subvariety Y ⊂ X we associate a canonical element of the Brauer group eY ∈ Br(X/Y ) and consider the derived category Db(X/Y, eY ) of modules over the corresponding Azumaya algebra on X/Y . It turns out that for every pair of lagrangian subvarieties in X there is an equivalence between these categories generalizing the Fourier-Mukai transform. The class eY is trivial if and only if the projection X → X/Y splits, in this case Db(X/Y, eY ) ≃ Db(X/Y ). This implies the ”if” part of the following conjecture: the derived categories of coherent sheaves on abelian varieties A and A are equivalent if and only if there is an isomorphism f : Â × A→̃Â′ × A such that (f × f)LA′ ≃ LA. In particular, for any abelian variety A and a symmetric homomorphism f : A →  we construct an equivalence Db(A) ≃ Db(A/ ker(fn)) where fn = f |An provided that mn ker(f) = 0 for some m relatively prime to n. The construction is based on analogy with the classical theory of representations of the Heisenberg group of a symplectic vector space: the categories Db(X/Y, eY ) are just different models of the same ”irreducible” representation of theHeisenberg groupoid—amonoidal groupoid naturally attached to (X,L). The corresponding analogue of Weil representation is studied in [4].