A Fourier-Mukai Transform for Stable Bundles on K3 Surfaces

We define a Fourier-Mukai transform for sheaves on K3 surfaces over \C, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli space of stable sheaves on $X$. For a wide class of K3 surfaces $\hat X$ can be chosen to be isomorphic to $X$; then the Fourier-Mukai transform is invertible, and the image of a zero-degree stable bundle $F$ is stable and has the same Euler characteristic as $F$.Comment: Revised version, 15 pages AMSTeX with AMSppt.sty v. 2.1

Fourier Mukai Transforms for Gorenstein Schemes

We extend to singular schemes with Gorenstein singularities or fibered in schemes of that kind Bondal and Orlov's criterion for an integral functor to be fully faithful. We also contemplate a criterion for equivalence. We offer a proof that is new even if we restrict to the smooth case. In addition, we prove that for locally projective Gorenstein morphisms, a relative integral functor is fully faithful if and only if its restriction to each fibre also is it. These results imply the invertibility of the usual relative Fourier-Mukai transform for an elliptic fibration as a direct corollary.Comment: Final version. To appear in Advances in Mathematic

Introduction to Fourier-Mukai and Nahm transforms with an application to coherent systems on elliptic curves

These notes record, in a slightly expanded way, the lectures given by the first two authors at the College on Moduli Spaces of Vector Bundles that took place at CIMAT in Guanajuato, Mexico, from November 27th to December 8th, 2006. The college, together with the ensuing conference on the same topic, was held in occasion of Peter Newstead's 65th anniversary. It has been a great pleasure and a privilege to contribute to celebrate Peter's outstanding achievements in algebraic geometry and his lifelong dedication to the progress of mathematical knowledge. We warmly thank the organizers of the college and conference for inviting us, thus allowing us to participate in Peter's celebration. The main emphasis in these notes is on the Fourier-Mukai transforms as equivalences of derived categories of coherent sheaves on algebraic varieties. For this reason, the first Section is devoted to a basic (but we hope, understandable) introduction to derived categories. In the second Section we develop the basic theory of Fourier-Mukai transforms. Another aim of our lectures was to outline the relations between Fourier-Mukai and Nahm transforms. This is the topic of Section 3. Finally, Section 4 is devoted to the application of the theory of Fourier-Mukai transforms to the study of coherent systems. This is a review paper. Most of the material is taken from [BBH08] and [HT08], although the presentation is different in some places

Cech and de Rham Cohomology of Integral Forms

We present a study on the integral forms and their Cech/de Rham cohomology. We analyze the problem from a general perspective of sheaf theory and we explore examples in superprojective manifolds. Integral forms are fundamental in the theory of integration in supermanifolds. One can define the integral forms introducing a new sheaf containing, among other objects, the new basic forms delta(dtheta) where the symbol delta has the usual formal properties of Dirac's delta distribution and acts on functions and forms as a Dirac measure. They satisfy in addition some new relations on the sheaf. It turns out that the enlarged sheaf of integral and "ordinary" superforms contains also forms of "negative degree" and, moreover, due to the additional relations introduced, its cohomology is, in a non trivial way, different from the usual superform cohomology.Comment: 20 pages, LaTeX, we expanded the introduction, we add a complete analysis of the cohomology and we derive a new duality between cohomology group

On a Conjecture About Higgs Bundles and Some Inequalities

We briefly review an open conjecture about Higgs bundles that are semistable after pulling back to any curve, and prove it in the rank 2 case. We also prove some results in higher rank under suitable additional assumptions. Moreover, we establish a set of inequalities holding for H-nef Higgs bundles that generalize some of the Fulton–Lazarsfeld inequalities for numerically effective vector bundles

The supermoduli of SUSY curves with Ramond punctures

We construct local and global moduli spaces of supersymmetric curves with Ramond-Ramond punctures. We assume that the underlying ordinary algebraic curves have a level n structure and build these supermoduli spaces as algebraic superspaces, i.e., quotients of étale equivalence relations between superschemes