11:45-12:45 O' Grady
Title: Bridgeland stability on Kuznetsov components in families
Abstract: Kuznetsov (based on earlier work by Bondal, Orlov and others) showed that frequently, the most interesting part of the derived category of a Fano variety X is a component of a certain semi-orthogonal decomposition that we call "Kuznetsov component" of X. In particular, the most interesting part of the geometry of moduli space of sheaves on X is induced by the Kuznetsov component.
I will explain a construction of stability conditions on the Kuznetsov component of many Fano 3-folds, and on the cubic fourfold. Combined with a notion of a family stability conditions on a family of varieties, this has many potential applications, some of which I will explain.
This is based on joint work (partly in progress) with Marti Lahoz, Emanuele Macri, Howard Nuer, Paolo Stellari and Alex Perry.
Title: Gorenstein Rings and Euler Stability
Abstract: A stability condition stratifies the unstable complexes of vector bundles according to their Harder-Narashimhan filtrations. There is a distinguished Euler stability condition on projective space that we apply to the minimal free resolutions of Gorenstein rings to obtain a new stratification of the space of symmetric tensors. This is joint work in progress with Brooke Ullery.
Title: Donaldson-Thomas invariants of the banana manifold and elliptic genera.
Abstract: The Banana manifold (or bananafold for short), is a compact Calabi-Yau threefold X which fibers over P1 with Abelian surface fibers. It has 12 singular fibers which are non-normal toric surfaces whose torus invariant curves are a banana configuration: three P1’s joined at two points, each of which locally look like the coordinate axes in C3. We show that the Donaldson-Thomas partition function of X (for curve classes in the fibers) has an explicit product formula which, after a change of variables is the same as the generating function for the equivariant elliptic genera of Hilb(C2), the Hilbert scheme of points in the plane. Implications for genus 0 Gopakumar-Vafa invariants will also be discussed.
Title: Derived categories of moduli spaces of stable rational curves
Abstract: I will report on joint work with Jenia Tevelev on Orlov's question about exceptional collections on moduli spaces of pointed stable rational curves.
Title: Geometry of Moduli of Abelian Differentials
Abstract: The study of moduli spaces of Abelian differentials with a given type of zeros has connections to many areas of mathematics. In this lecture we will introduce this topic from the perspective of algebraic geometry, with a focus on recent developments and open problems.
Title: The cohomology and birational geometry of moduli spaces of sheaves on surfaces
Abstract: In the first half of this talk, I will discuss joint work with Jack Huizenga on computing the cohomology of the general sheaf in moduli spaces of stable sheaves on rational surfaces. This generalizes a celebrated theorem of Göttsche and Hirschowitz. As a consequence, we classify Chern characters on Hirzebruch surfaces such that the general bundle with that character is globally generated. In the second half, I will describe joint work with Matthew Woolf on the stabilization of the cohomology of moduli spaces of sheaves on surfaces.
Title: Infinitesimal deformations of log Calabi Yau varieties and orbifolds
Abstract: A Calabi Yau complex projective variety has unobstructed deformations by the theorem of Tian-Todorov; a similar result holds in the smooth log case, i.e. for deformation of a pair (X,D) where X is a smooth projective variety and D a smooth anti canonical divisor. In joint work with Donatella Iacono, we show how to extend this result to orbifolds and discuss the case where D had simple normal crossing.
Title: Stability of Hilbert points and applications
Abstract: A Hilbert point of a projective variety records a piece of its homogeneous ideal in some embedding into a projective space. Geometric invariant theory of Hilbert points has been classically used in constructing moduli spaces of polarized varieties by Mumford, Gieseker, and many others. Stability of small (as opposed to asymptotic) Hilbert points is much less explored. In this talk, I will discuss stability analyses for small Hilbert points (and their generalizations, syzygy points) of canonical curves, certain K3 and del Pezzo surfaces, and Gorenstein Artin algebras. Applications will include new constructions in the Hassett-Keel program for the moduli space of curves and an invariant-theoretic Mather-Yau theorem.
Title: Rationality in families
Abstract: Recent work of Voisin, Colliot-Th'el`ene, Pirutka, Totaro, and others shows that the technique of decomposition of the diagonal has broad
implications for rationality problems. We now know that large classes of rationally connected threefolds generally fail to be stably rational and
that rationality is not a deformation invariant of smooth projective fourfolds. This gives new impetus to the long-standing question of
characterizing the locus of rational members in a family of smooth projective varieties. We discuss recent progress for several key examples.
Title: Degenerations of Hilbert schemes of degree 0 cycles on surfaces
Abstract: In this talk I will report on a GIT construction for degenerations of Hilbert schemes of points on surfaces. The motivation comes from studying degenerations of irreducible holomorphic symplectic manifolds (IHMS), where a prime examples is given by degree n Hilbert schemes of K3 surfaces. Previously, Nagai gave a very concrete construction for degree 2 Hilbert schemes by using ad hoc modifications of the relative Hilbert scheme. In contrast to that, Li and Wu have developed a very general theory of degenerations of Hilbert schemes (not only of 0-cycles) using expanded degenerations. Being very general, this approach makes it hard to describe the degenerations and their geometry explicitly. Here we develop a third approach: we use GIT methods to construct degenerations of Hilbert schemes of points on surfaces (in arbitrary degree). This allows us to describe the geometry of the singular fibres very explicitly. We can further prove that, in the case of degenerations of K3 surfaces, this is leads to dlt-degenrations of the Hilbert schemes and by work of Halle and Nicaise this shows that the dual complex of the degenerate fibre coincides with the Kontsevich-Soibelman skeleton. There is also a relationship with recent work by Kollár, Laza, Saccà and Voisin. This is joint work with M. Gulbrandsen, L. Halle and (partly) Z. Zhang.
Title: On the possible Betti tables of a canonical curve
Abstract: A famous conjecture of Green (now a theorem of Voisin) completely describes the Betti table of a canonical curve of maximal gonality. More recently, Schreyer has asked what one can say about the possible Betti tables of curves in the case where the gonality is non-maximal. In particular, it seems that the last entry of the first row contains some very fine geometric information. We will discuss several results, including a proof of Schreyer's conjecture and some broad generalisations of it. This is joint work with G. Farkas.
Title: Applications of non-reductive geometric invariant theory
Abstract: In general geometric invariant theory (GIT) for non-reductive linear algebraic group actions is much less well behaved than for reductive actions. However when the unipotent radical U of a linear algebraic group H is graded, in the sense that a Levi subgroup has a central one-parameter subgroup which acts by conjugation on U with all weights strictly positive, then GIT for a linear action of H on a projective scheme is almost as well behaved as in the reductive setting, provided that we are willing to multiply the linearisation by an appropriate rational character. This has potential applications for the construction of moduli spaces of 'unstable' objects of prescribed type, such as sheaves of fixed Harder-Narasimhan type or unstable curves.
Title: Bridgeland stability and the genus of space curves
Abstract: I will give an introduction to various notions of stability in the bounded derived category of coherent sheaves on the three-dimensional projective space.
As application I will show how to use these techniques towards the study of space curves.
This is joint work in progress with Benjamin Schmidt.
Title: Abelian varieties associated to hyperkählers of Kummer type.
Title: Monodromy and derived equivalences
Abstract: This will be a report on our joint work with Roman Bezrukavnikov in which we prove that derived automorphisms of the Hilbert scheme of points in the plane (and many other symplectic resolutions) which were constructed by Bezrukavnikov and Kaledin using quantization in characteristic p>>0 categorify the monodromy of the quantum differential equation.
Title: Polynomiality of the double ramification cycle
Abstract: The double ramification cycle parametrizes curves admitting maps to the projective line with specified ramification profiles over two points. In the first half of this talk, I will describe joint work with Don Zagier giving a new formula for this cycle (in terms of generalized boundary strata classes) that is visibly polynomial in the parts of the ramification profiles. In the second half, I will discuss joint work with Georg Oberdieck applying this polynomiality to prove new results in the Gromov-Witten theory of an elliptic curve.
Title: Cubic fourfolds, hyper-Kähler manifolds and their degenerations
Abstract: There at least three families of hyper-Kähler manifolds built from cubic fourfolds, the most recently discovered one being the compactified intermediate Jacobian fibrations I constructed with Laza and Saccà. In a joint work with Kollár, Laza and Saccà, we provide an easy way to compute their deformation types, by proving that if the central fiber of a degeneration of hyper-Kähler manifolds
has one component which is not uniruled, then after base-change the family becomes fiberwise birational to a family of smooth hyper-Kähler manifolds.