Schedule and abstracts

Abstract: Let S be a polarized K3 surface, and let P(x) be a Hilbert polynomial.  The moduli stack M of semistable coherent sheaves (with respect to the polarization) with Hilbert polynomial P(x) can be very singular.  Daniel Halpern-Leistner conjectured that the natural mixed Hodge structure on this stack is nonetheless pure, and I'll explain the proof of this conjecture.  Using recent results (joint work with Lucien Hennecart and Sebastian Schlegel Mejia) I'll explain how the ideas in the proof yield a lot more, in particular: extensions of Halpern-Leistner's result, stating that for primitive P(x) the Poincaré polynomial of the stack is independent of the (sufficiently generic) polarization to general P(x), as well as a version for intersection Poincaré polynomials.

Abstract: In this joint work with D. Mattei and Y. Prieto we showed that a general projective Hyperkähler manifold that is deformation equivalent to the Hilbert scheme of n-points on a K3 surface (i.e., of K3[n]-type) cannot admit non-trivial birational self-maps of finite order. More precisely, we showed that whenever such maps exist, the manifold must be a moduli space of sheaves on K3 surfaces. This prompted us to investigate birational self-maps on these moduli spaces. Using Markman’s theory of hyperkähler lattices and Bayer–Macri’s study of Bridgeland stability on K3 surfaces, we imposed explicit numerical constraints on the topological invariants of the sheaves so that certain birational involutions exist on their moduli space.

Abstract: I will talk about work with Hacking, Keel and Siebert on using mirror constructions to provide partial compactifications of the moduli of K3 surfaces. Starting with a one-parameter maximally unipotent degeneration of Picard rank 19 K3 surfaces, we construct, using methods of myself and Siebert, a mirror family which is defined in a formal neighbourhood of a union of strata of a toric variety whose fan is defined, to first approximation, as the Mori fan of the original degeneration. This formal family may then be glued in to the moduli space of polarized K3-surfaces to obtain a partial compactification. Perhaps the most significant by-product of this construction is the existence of theta functions in this formal neighbourhood, certain canonical bases for sections of powers of the polarizing line bundle.

Abstract: The goal of this talk is to discuss the classification problem of spherical objects in various geometric settings including minimal resolution of du Val singularities and 3-fold flopping contractions. Spherical objects are objects in the derived category that can provide an autoequivalence as twist functors, but the main theorem can classify more general objects. We give a classification theorem of all spherical-like objects, which will be called brick complexes, in the null category, which is the kernel of the derived global section. The key technique of the proof comes from an analogy with the theory of silting-discrete algebras. The same technique goes further, and gives a classification of all bounded t-structures in the null category. As a corollary, classifications give results on Bridgeland stability conditions and the autoequivalence group.

Abstract: Let X be a smooth projective Fano threefold with Picard number one. When its index is not 1, or it is with index 1 and genus greater than 5, the Kuznetsov component of X, Ku(X), admits a stability condition in the sense of Bridgeland. The moduli space of stable objects in Ku(X) behaves similarly to that of the stable sheaves on smooth projective curves. I will talk about some results that we can show by the time of the workshop. This is an ongoing joint work with Laura Pertusi and Xiaolei Zhao.

Abstract: Gromov--Witten (GW) invariants of genus g, with g greater than one, do not count curves of genus g in a given space: curves of lower genus also contribute to GW invariants. In genus one this problem was corrected by Vakil and Zinger, who defined more enumerative numbers called "reduced GW invariants". More recently Hu, Li and Niu gave a construction of reduced GW invariants in genus two. I will define reduced Gromov--Witten invariants in all genera. This is work with A. Cobos-Rabano, E. Mann and R. Picciotto.

Abstract: A cohomology class of a smooth complex variety of dimension n is said to be of coniveau at least c if it vanishes on the complement of a closed subvariety of codimension at least c, and of strong coniveau at least c if it comes about by proper pushforward from the cohomology of a smooth variety of dimension at most n–c. The notions of coniveau and strong coniveau each define a filtration on the cohomology groups of a variety, and these filtrations are known to coincide in many cases. In the talk, I will explain a construction of some new examples where the filtrations differ, which are found in joint work with Jørgen Vold Rennemo.

Abstract: We study the derived category of singular varieties with at most quotient singularities. Particularly, we study the derived category of varieties with 1/n(1,...,1) singularities. We give sufficient conditions for when derived categories of such varieties admit a semiorthogonal decomposition into two components; a "small" component containing the information of the singularity and a "big" component encoding the smooth information. We give various examples satisfying our condition, and we also give criteria which do not allow such decompositions. This is joint work in progress with M. Kalck and Y. Kawamata.

Abstract: It is by now well-known that there are many exact functors between derived categories of coherent sheaves on smooth projective varieties that do not admit a lift between corresponding DG enhancements. While the structure of a triangulated functor is insufficient for many constructions in algebraic geometry, it is actually possible to add additional structure to these non-enhanceable functors if one considers the formalism of A_n categories, a truncated version of the A-infinity category axioms.

Abstract: We consider a mutation-finite quiver Q. Associated to Q are two interesting complex manifolds: 1) the space of stability conditions for the derived category of the Ginzburg algebra associated to the quiver, 2) the complex cluster Poisson variety. Each of these manifolds is constructed from the combinatorics of the quiver, though in very different ways.

We introduce an "interpolation" of these complex manifolds: we give a construction of a complex manifold together with a submersion to the complex plane which we call the stability twistor space. The fiber over the origin is a finite quotient of the space of stability conditions whereas every other fiber is an etale cover of a quotient of the cluster Poisson variety associated to the quiver. This ongoing project is joint with Tom Bridgeland.

Abstract: Moduli spaces of algebraic K3 surfaces admit several natural compactifications, amongst which are the KSBA and toroidal compactifications. KSBA compactifications are higher dimensional generalisations of the Deligne-Mumford compactification of the moduli space of curves. They have the compelling advantage that the boundary is "modular"; in other words, points on the boundary provide moduli for a class of stable surfaces with controlled singularites. However, KSBA compactifications are also typically very difficult to explicitly describe. On the other hand, toroidal compactifications are compactifications constructed via a choice of combinatorial data, in the form of several fans. They are beautifully explicit, but there is not a unique choice for the defining fans and it is typically very difficult to give a modular interpretation for their boundaries. I will present joint work with Alexeev and Engel, in which we describe an explicit relationship between the KSBA compactification and a particular toroidal compactification in the case of K3 surfaces of degree 2, and I will describe how this work has since been generalised by Alexeev and Engel to wider classes of K3 surfaces.

Abstract:  Abelian varieties over the complex numbers appear as intermediate Jacobians of smooth projective varieties. The classical example is the Albanese variety, and the Abel-Jacobi map in this case is the Albanese map on 0-cycles. Another sort of example is given by  the intermediate Jacobian built on degree 3 cohomology of a rationally connected variety, which is the target of the Abel-Jacobi map for codimension 2 cycles. We discuss the notion of universal cycle in relation with rationality questions and show that, for  a rationally connected threefold, the existence of a universal codimension 2 cycle can always be reduced to the existence of a universal 0-cycle for a smooth projective surface.  Focusing on  the case of 0-cycles, we consider the problem of the existence of a universal 0-cycle for Brauer-Severi varieties or rationally connected fibrations over abelian varieties and relate this problem to various cases of the integral Hodge conjecture for abelian varieties.