Title: Enriques varieties
Abstract: Enriques varieties are defined as a higher dimensional generalization of Enriques surfaces. In this talk, I'll discuss some of their properties and construct examples using fixed point free automorphisms on generalized Kummer varieties.
Title: Hall algebras and curve-counting invariants
Abstract: I will start by explaining how Dominic Joyce used relative Grothendieck rings of algebraic stacks to define Hall algebras of coherent sheaves. I will then try to explain how such algebras can be used to prove results about Donaldson-Thomas invariants of Calabi-Yau threefolds.
Title : Gromov--Witten Invariants and Modular Forms.
Title : Extremal Laurent Polynomials, Picard-Fuchs equations, and Fano 3-folds
Abstract: The talk is based on experimental work in progress with (at least) T. Coates, V. Golyshev and A. Kasprzyk. Following Golyshev, a Laurent polynomial f is extremal if the 'main' piece of Rf! is a local system of smallest ramification. I describe a program to classify ELP supported on reflexive polytopes by computer. Conjecturally, ELP correspond under mirror symmetry to 'pieces' of quantum cohomology of Fano manifolds--that is, 'quantum motives.' I discuss this interpretation in the case of ELPs supported on reflexive polytopes in 3 dimensions.
Title: Reductions of irreducible symplectic varieties defined over number fields
Abstract : Let X be an irreducible symplectic variety defined over a number field K. We prove that if X has second Betti number at least 4 then there exist a finite algebraic field extension L/K and a density 1 set S of non-archimedean places of L such that the reduction of X at any place in S has nonzero Hasse-Witt invariant.
Title: Deformation Rigidity Of Holomorphic Maps Onto Some Toric Manifolds.
Abstract: We shall show that for a holomorphic map f: Y -> X onto a "complicated" toric projective manifold X, every deformation of f comes from the automorphisms of X. In particular, if Y is toric, then f is necessarily toric.
Title: On a conjecture of Beltrametti and Sommese.
Abstract: Let X be a projective manifold of dimension n. Beltrametti and Sommese conjectured that if A is an ample divisor such that K+(n-1)A is nef, then K+(n-1)A has non-zero global sections. We prove a generalised version of this conjecture in arbitrary dimension. In dimension three, we prove the stronger non-vanishing conjecture of Ambro, Ionescu and Kawamata and give an application to Seshadri constants.
Title: Stability conditions and spherical objects on K3 surfaces
Abstract: I will discuss stability conditions on K3 surfaces from the point of view of spherical objects. By work of Bridgeland this is related to the group of autoequivalences. His beautiful conjecture, which remains open, describes the latter as a (orbifold) fundamental group of a quasi-projective variety.
Title: Hodge structure on cyclic homology: results and conjectures
Abstract: Periodic cyclic homology is a generalization of de Rham cohomology to non-commutative algebras and DG algebras. As it happens, some of the usual properties of de Rham cohomology of algebraic varieties survive in this generalization: for example, one can prove a version of the Hodge-to-de Rham degeneration theorem. In char p, one can further show that periodic cyclic homology carries a structure of a "filtered Dieudonn\'e module" of Fontaine-Lafaille, a p-adic analog of a mixed Hodge structure. One would expect that in char 0, periodic cyclic homology carries a Hodge structure in the usual sense. This is quite far from being established; however, we do have some results. I will give a brief overview of what one expects and what one can actually prove.
Title: On numerical Zariski decomposition
Abstract: I will talk about a numerical version of the Zariski decomposition which is easier to handle as examples of applications show.
Title: Differential Forms on Singular Spaces.
Abstract: The talk is concerned with differential forms on log canonical varieties. It is shown that any p-form defined on the smooth locus of a variety with canonical or klt singularities extends regularly to any resolution of singularities. In fact, a much more general theorem for log canonical pairs is established. The proof relies on vanishing theorems for log canonical varieties and on methods of the minimal model program. In addition, a theory of differential forms on dlt pairs is developed. It is shown that many of the fundamental theorems and techniques known for sheaves of logarithmic differentials on smooth varieties also hold in the dlt setting.
Immediate applications include the existence of a pull-back map for reflexive differentials, generalisations of Bogomolov-Sommese type vanishing results, and a positive answer to the Lipman-Zariski conjecture for klt spaces. Relations to moduli theory are emphasized.
Title: Slodowy slices and symplectic hypersurfaces.
Title: Cubic hypersurfaces of dimension seven and their derived category
Abstract: Kuznetsov has proved that the derived category of a cubic sevenfold contains a three-dimensional Calabi-Yau subcategory. I will show how one can try to understand this Calabi-Yau category from representations of the cubic as a linear section of a cubic hypersurface of dimension 25, invariant under the exceptional Lie group E6.
Title: The Beauville-Bogomolov class as a characteristic class.
Abstract: Let X be any compact Kahler manifold deformation equivalent to the Hilbert scheme of length n subschemes on a K3 surface, n>1. For each point x in X we construct a rank 2n-2 reflexive coherent twisted sheaf E on X, locally free away from the point x, with the following properties.
Title: Fano models of Enriques surfaces and associated symplectic 4-folds
Title: Poisson deformations of symplectic varieties and associated Galois covers.
Title: Complex manifolds homotopically equivalent to tori.
Abstract: I discuss joint recent work with F.Catanese and K.Oguiso. It is shown that a compact complex threefold with trivial canonical bundle admitting a non-constant meromorphic function (or a non-trivial map to a compact complex surface) which is homotopically equivalent to a complex torus, is actually biholomorphic to a torus.
Title: Hirzebruch-Riemann-Roch formula for matrix factorizations.
Abstract: This is a joint work with A.Vaintrob. We compute explicitly the ingredients of the Shklyarov's categorical HRR formula in the case of the category of matrix factorizations associated with an isolated singularity.
Title: Derivative complex, BGG correspondence, and cohomology.
Abstract: I will describe joint work with R. Lazarsfeld in which we use various incarnations of the derivative complex, together with recent results on generic vanishing, in order to provide new results on the cohomology of compact Kaehler manifolds. By linking it with the BGG correspondence, we show that the cohomology of the canonical bundle has a surprisingly simple structure as a module over the exterior algebra, while by linking it to the theory of bundles on projective spaces, we obtain numerical inequalities for the Hodge numbers and holomorphic Euler characteristic.
Title: A tropical view on Landau-Ginzburg models - toward Homological Mirror Symmetry for Fano varieties.
Title: Fano varieties of lines of cubic fourfolds containing a plane.
Abstract: An intriguing idea, going back to Bondal and Orlov, is that, for a smooth projective variety, the bounded derived category of coherent sheaves (and semi-orthogonal decompositions of it) should encode interesting information about the geometry of the variety itself. This belief will be studied in the case of smooth cubic hypersurfaces of dimension 4 and 3. In particular, we will consider the Fano varieties of lines of cubic 4-folds containing a plane. We will show that, for generic 4-folds of this type, the Fano variety is isomorphic to a moduli space of twisted stable complexes on a K3 surface. On the other hand, we will prove that those Fano varieties are always birational to moduli spaces of twisted stable coherent sheaves on a K3 surface. If time permits, we will discuss other results concerning cubic 3-folds. This is joint work with Emanuele Macri.