arXiv subjects
Publications and source records attributed to Oren Ben-Bassat.
We give an explicit groupoid presentation of certain stacks of vector bundles on formal neighborhoods of rational curves inside algebraic surfaces. The presentation involves a Möbius type action of an automorphism group on a space of extensions.
We show that any complex manifold that has a divisor whose normalization has non-zero first Betti number either has a non-trivial holomorphic gerbe which does not trivialize meromorphicly or a meromorphic line bundle not equivalent to any holomorphic line bundle. Similarly, higher Betti numbers of divisors correspond to higher gerbes or meromorphic gerbes. We give several new examples.
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. This represents a modified version of my May 2006 thesis for the University of Pennsylvania. The only addition is the subsection titled 'Alternative Method'.
The classical Fourier-Mukai duality establishes an equivalence of categories between the derived categories of sheaves on dual complex tori. In this article we show that this equivalence extends to an equivalence between two dual objects. Both of these are generalized deformations of the complex tori. In one case, a complex torus is deformed formally in a non-commutative direction specified by a holomorphic Poisson structure. In the other, the dual complex torus is deformed in a B-field direction to a formal gerbe. We show these two deformations are Fourier-Mukai equivalent.
In this paper we develop a relative version of T-duality in generalized complex geometry which we propose as a manifestation of mirror symmetry. Let M be an n-dimensional smooth real manifold, V a rank n real vector bundle on M, and nabla a flat connection on V. We define the notion of a nabla-semi-flat generalized complex structure on the total space of V. We show that there is an explicit bijective correspondence between nabla-semi-flat generalized complex structures on the total space of V and nabla(dual)-semi-flat generalized complex structures on the total space of the dual of V. Similarly we define semi-flat generalized complex structures on real n-torus bundles with section over an n-dimensional base and establish a similar bijective correspondence between semi-flat generalized complex structures on pair of dual torus bundles. Along the way, we give methods of constructing generalized complex structures on the total spaces of vector bundles and torus bundles with sections. We also show that semi-flat generalized complex structures give rise to a pair of transverse Dirac structures on the base manifold. We give interpretations of these results in terms of relationships between the cohomology of torus bundles and their duals. We also study the ways in which our results generalize some well established aspects of mirror symmetry as well as some recent proposals relating generalized complex geometry to string theory.
The main goal of our paper is the study of several classes of submanifolds of generalized complex manifolds. Along with the generalized complex submanifolds defined by Gualtieri and Hitchin (we call these ``generalized Lagrangian submanifolds'' in our paper), we introduce and study three other classes of submanifolds. For generalized complex manifolds that arise from complex (resp., symplectic) manifolds, all three classes specialize to complex (resp., symplectic) submanifolds. In general, however, all three classes are distinct. We discuss some interesting features of our theory of submanifolds, and illustrate them with a few nontrivial examples. We then support our ``symplectic/Lagrangian viewpoint'' on the submanifolds introduced by Gualtieri and Hitchin by defining the ``generalized complex category'', modelled on the constructions of Guillemin-Sternberg and Weinstein. We argue that our approach may be useful for the quantization of generalized complex manifolds.