Twisting derived equivalences

Abstract

Abstract. We introduce a new method for "twisting" relative equivalences of derived categories of sheaves on two spaces over the same base. The first aspect of this is that the derived categories of sheaves on the spaces are twisted. They become derived categories of sheaves on gerbes living over spaces that are locally (on the base) isomorphic to the original spaces. Secondly, this is done in a compatible way so that the equivalence is maintained. We apply this method by proving the conjectures of Donagi and Pantev on dualities between gerbes on genus-one fibrations and comment on other applications to families of higher genus curves. We also include a related conjecture in Mirror Symmetry.

Vector bundles on non-Kaehler elliptic principal bundles

“…In the third and fourth sections, using the relative Jacobian, we adapt the construction of Cȃldȃraru, [Cȃl02], to our case, obtaining a twisted Fourier-Mukai transform. Similar results were obtained in different settings by O.Ben-Bassat [BB09] and I.Burban and B.Kreussler [BK06]. In the fifths section using this transform and the associated spectral cover we prove that the moduli space of rank n, relatively semi-stable vector bundles is corepresented by the relative Douady space of length n and relative dimension 0 subspaces of the relative Jacobian, see theorem 5.…”

Langlands duality for Hitchin systems

“…In general, the duality between a family of abelian varieties A → B over a base B and its dual family A ∨ → B is given by a Poincare sheaf which induces a Fourier-Mukai equivalence of derived categories. It is well known [DP08,BeBra07,BB09] that the Fourier-Mukai transform of an A-torsor A α is an O * -gerbe α A ∨ on A ∨ . In our case there is indeed a natural stack mapping to Higgs, namely the moduli stack Higgs ss of semistable G-Higgs bundles on C. Over the locus of stable bundles, the stabilizers of this stack are isomorphic to the center Z(G) of G and so over the stable locus Higgs ss is a gerbe.…”

“…The above tables and the calculation of the cocharacter lattices of the Prym varieties P (7) with the duality for representations of commutative group stacks described in Arinkin's appendix to [DP08] (see also [BeBra07]), or invoke the recent result [BB09] of O.Ben-Bassat. In fact, Ben-Bassat's proof works in a much more general context and will imply the full categorical duality even over the discriminant ∆, as long as one can show that the Poincare sheaf on the cameral Pryms extends across ∆.…”

Kernel algebras and generalized Fourier–Mukai transforms

“…This is a very natural device for producing gerbes on X=G, similar to gerbe data of [16] (resp. presentation of a gerbe of [10]). Namely, an action of G on X gives rise to the so called action groupoid G X G X over X, where G consists of .x; g; x 0 / such that x D gx 0 .…”

“…The relevant notion of a 1-cocycle on a groupoid generalizes the well known description of gerbes using open coverings and line bundles on pairwise intersections (see Sec. 1 of [16], and also [31], [10]). …”