VBAC2015
(Vector Bundles on Algebraic Curves 2015)

TBA

The goal of this talk is to explain how to extend Grassmannians to the world of derived stacks, i.e. how to construct a satisfying derived enhancement of Grassmannian varieties. I will begin by discussing some background about derived geometric stacks and in particular I will focus on Artin-Lurie-Pridham representability theorem, which provides us with a "computational" criterion to check whether a simplicial presheaf of derived algebras is a derived geometric stack. Then I will use such a result in order to study derived moduli of perfect complexes and filtered perfect complexes over a base scheme; finally the derived Grassmannian will arise as some suitable homotopy limit of such stacks. Time permitting I will end by sketching how to use this derived version of the Grassmannian to obtain a derived version of Griffiths' period mapping.

Let be an abelian scheme over a scheme . The Fourier-Mukai transform gives an equivalence between the derived category of and the derived category of the dual abelian scheme. We partially extend this to certain schemes over (which we call degenerate abelian schemes) whose generic fiber is an abelian variety, while special fibers are singular. Our main result provides a fully faithful functor from a twist of the derived category of to the derived category of . Here is the algebraic space classifying fiberwise numerically trivial line bundles.

Next, we show that every algebraically integrable system gives rise to a degenerate abelian scheme and discuss applications to Hitchin systems.

The existence-uniqueness problems of Fourier-Mukai kernels of exact functors are studied investigating the properties of the functor which maps to the Fourier-Mukai functor . Dg-enhancements, together with Toën’s theorem allow us to address the problem in purely algebraic terms, studying the properties of the functor where , , are pretriangulated dg-categories. The dg-category . is the internal Hom in the homotopy category of dg-categories; elements of . are called quasi-functors. In our work, we concentrated on the the dg-lift uniqueness problem, which is equivalent to the problem of uniqueness of Fourier-Mukai kernels. Namely: is functor essentially injective? That is, given quasi-functors , , : , is it true that implies ? The problem is, to my knowledge, difficult in general. In the talk I will explain some attempts which give at least some partial answers. The techniques employed involve results in the homotopy theory of dg-categories, and the gluing construction of two dg-categories along a bimodule.

Donaldson–Thomas invariants are virtual counts of (stable) coherent sheaves over Calabi–Yau threefolds, attached to the symmetric perfect obstruction theory possessed by their moduli spaces. The definition of DT invariants makes sense also over abelian threefolds, but they are almost always zero. This is, however, due to the appearence of the Euler characteristics of the abelian threefold and its dual as factors, and the "remaining factor" is meaningful and nontrivial. The resulting invariants can be viewed as virtual counts for the orbits of the identity component of the group of autoequivalences of the derived category.

In this talk we will examine the question of whether derived equivalent (smooth, projective) varieties over finite fields have equal zeta functions, which is closely related to Orlov's conjecture that such varieties should have isomorphic Chow motives with rational coefficients. We will show some positive results.

We give a new construction of algebraic symplectic varieties by taking a non-reductive algebraic symplectic reduction of the cotangent lift of an action of the additive group on an affine space. For a linear action of the additive group on an affine space over the complex numbers, the non-reductive GIT quotient is isomorphic to a reductive affine GIT quotient; however, we show that the corresponding non-reductive and reductive algebraic symplectic reductions are not isomorphic, but rather birationally symplectomorphic. If time permits, we will discuss work in progress on non-linear actions.

Let be a complete DVR with fraction field of characteristic 0 and algebraically closed residue field of characteristic . Let . be a smooth, projective morphism with geometrically integral fibres and . Denote by a line bundle on . By extending a result of V. Artamkin, we show that the moduli space of Gieseker stable sheaves on the surface . with determinant is smooth over .

This is a survey on some aspects of Fourier-Mukai Transforms. After a fairly detailed description of Mukai's results I will present some generalizations and improvements due to Orlov and others. As in Mukai's work the emphasis will lie on applications to abelian varieties and K3-surfaces. If time allows, I will also recall some applications to birational geometry.

We will prove that, as happens for smooth elliptic curves, any Fourier-Mukai partner of a projective reduced Gorenstein curve of genus one and trivial dualizing sheaf is isomorphic to itself. The result is obtained by using powerful tools of the theory of Fourier-Mukai functors for singular schemes. We will also discuss the potential application of these ideas to the study of Fourier-Mukai partners of higher dimensional elliptic fibrations.

Following on from Part I, we will look at the developments of Fourier-Mukai transforms to other varieties and spaces. We will also look at the role of Fourier-Mukai transforms in studying moduli spaces of sheaves and their generalizations.

Among abelian varieties, Jacobians of smooth projective curves C have the important property of being autodual, i.e., they are canonically isomorphic to their dual abelian varieties. This is equivalent to the existence of a Poincaré line bundle on which is universal as a family of algebraically trivial line bundles on . Mukai’s results give, in our case, a yet other instance of this fact: the Fourier-Mukai transform with kernel P is an auto-equivalence of the bounded derived category of .

I will talk on joint work with Filippo Viviani and Antonio Rapagnetta, where we try to generalize both the autoduality result and Mukai’s equivalence result for singular reducible curves with locally planar singularities. Our results generalize previous work of Arinkin, Esteves, Gagné, Kleiman and can be seen as an instance of the geometric Langlands duality for the Hitchin fibration.

Let be a smooth projective curve over and the moduli space of vector bundles over of rank 2 and with fixed determinant of degree 1.Then the Fourier–Mukai functor from the bounded derived category of coherent sheaves on to that of given by the normalised Poincare bundle is fully faithful if the genus of is greater than or equal to six. The result is also true for curves of genus 2 and for non-hyperelliptic curves of genus 3, 4 and 5. This is proved by establishing precise vanishing theorems for a family of vector bundles on the moduli space . Results on the deformation and inversion of Picard bundles (already known) follow from the fully faithfulness of the FM functor.

TBA

Using the Poincare bundle as a kernel, sheaves on a smooth, projective curve become complexes on its Picard variety. There are some results when semistable bundle induce semistable Picard sheaves. Using Falting's homological characterisation of semistability, we address this question. This is a joint work with Georg Hein.

TBA

Jørgen Rennemo (Imperial): “The homological projective dual of  ”

In recent years, some powerful tools for computing semi-orthogonal decompositions of derived categories of algebraic varieties have been developed: Kuznetsov's theory of homological projective duality and a closely related technique of variation of GIT quotients for LG models.

I will explain how the latter works and how it can be used to understand the derived categories of complete intersections in . We obtain a description of the homological projective dual of the stack as a category of modules over a certain sheaf of Clifford algebras. In certain cases we can interpret this category geometrically, and by doing this we obtain a new proof of a result of Hosono and Takagi, which says that a certain pair of non-birational Calabi-Yau 3-folds are derived equivalent.

Orlov’s famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier-Mukai functor. We will show that this result is false without the fully faithfulness hypothesis. This is joint work with Michel Van den Bergh.

We shall study ringed finite spaces and quasi-coherent modules on them. We shall first see how these structures appear in a natural way when studying locally constant sheaves on a topological space or quasi-coherent modules on a scheme. After reviewing basic facts of ringed finite spaces, we shall study those ringed finite spaces where a theory of integral functors for quasi-coherent modules may be developed. Then we shall introduce the notions of schematic finite spaces and schematic morphisms, and see that they behave, with respecto to quasi-coherence, like schemes and morphisms of schemes do. Finally, we shall compare the category of schematic finite spaces and schematic morphisms with the category of quasi-compact and quasi-separated schemes.i

It was a general belief and a formal conjecture by Bondal, Larsen and Lunts that the dg enhancement of the bounded derived category of coherent sheaves or the category of perfect complexes on a (quasi-)projective scheme is unique. This was proved by Lunts and Orlov in a seminal paper. In this talk we will explain how to extend Lunts-Orlov's results to several interesting geometric contexts. Namely, we care about the category of perfect complexes on noetherian separated schemes with enough locally free sheaves and the derived category of quasi-coherent sheaves on any scheme. This is a joint work in progress with A. Canonaco.

We outline the construction of an integral transform for certain singular solutions of Hitchin's equations on the complex projective line and discuss its properties. In particular we show that it gives a hyper-Kaehler isometry between moduli spaces. The proof uses non-Abelian Hodge theory and moduli spaces of sheaves on ruled Poisson surfaces.

We study the group of relative Fourier-Mukai transforms for Weierstrass fibrations, Fano or anti-Fano fibrations and abelian schemes. We describe this group completely in the first two cases. In the latter case we prove that if two abelian schemes are relative Fourier-Mukai partners then there is an isometric isomorphism between the fibre products of each of them and its dual abelian scheme. We show that these two conditions are equivalent if the base is normal and the slope map is surjective. Moreover in this situation we completely determine the group of relative Fourier-Mukai transforms and prove that the number of relative Fourier–Mukai partners of a given abelian scheme is finite.

For an abelian variety the moduli of line bundles is regarded as the dual of . Moreover is regarded as the moduli of line bundles on by the universal family. We shall explain that a similar statement holds for a K3 surface . If a fine moduli space of stable sheaves is a K3 surface, then can be regarded as a moduli space of stable sheaves on under suitable conditions. Since the result was proved several years ago (cf. arXiv:1003.2522), I will explain a slightly different proof based on the theory of Bridgeland stability conditions as in a paper of Huybrechts.