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Meeting ID: 899 9398 2042

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It is a classical fact, that for a family of surfaces over a smooth curve with smooth general fiber, if the special fiber has only rational double points as singularities, then after a possible finite base change, one can resolve the singularities of the total space and the special fiber by a common blowup. I will talk about a categorical version of this construction that also works in higher dimensions and about its applications.

https://ed-ac-uk.zoom.us/j/85159160108

Password: a simply-connected two-dimensional variety with trivial canonical bundle (omit the space).

I will describe a new correspondence between coherent sheaves on the projective plane and holomorphic curves in the projective plane with tangency condition along a smooth cubic curve. This correspondence is motivated by a combined application of hyperkähler rotation and mirror symmetry. The actual proof uses tropical geometry as a connecting bridge.

For a smooth cubic hypersurface Y, Sergey Galkin and Evgeny Shinder exhibited a relation between the naive motives of Y, the Fano variety F(Y) of lines and the Hilbert scheme Y^{[2]} of two points on Y. This relation has been shown to persist both on the level of rational Chow motives and integral Hodge structures. In a joint work with Pieter Belmans and Lie Fu, we lift this relation to derived categories by exhibiting a corresponding semi-orthogonal decomposition for the derived category of Y^{[2]}. I will explain how to obtain this semi-orthogonal decomposition from a refinement of Bondal-Orlov's results on derived categories of flips and how to further deduce an isomorphism of integral Chow motives using a recent result of Qingyuan Jiang.

Zoom link: https://ed-ac-uk.zoom.us/j/81407486222

Meeting ID: 814 0748 6222

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Consider the Quot scheme parametrising length n quotients of the rank r trivial bundle on C^4. There's a natural torus action on this scheme, for which the fix points are labelled by r-tuples of solid partitions of total size n. Nekrasov and Piazzalunga have assigned rational function weights to these fix points and conjectured a formula for the generating function of weighted counts of fix points. Oh and Thomas have defined general K-theoretic sheaf counting invariants for Calabi-Yau 4-folds and proved a torus localisation formula for these. We show that Nekrasov-Piazzalunga's weights agree with weights coming from the Oh-Thomas localisation formula (matching up the signs is the tricky part). We use this to prove that Nekrasov-Piazzalunga's conjectured formula for the generating function is correct. This is joint work with Martijn Kool.

Zoom link: https://ed-ac-uk.zoom.us/j/88687333719

Meeting ID: 886 8733 3719

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In this talk we study certain moduli spaces of semistable objects in the Kuznetsov component of a cubic fourfold. We show that they admit a symplectic resolution \tilde{M} which is a smooth projective hyperkaehler manifold deformation equivalent to the 10-dimensional example constructed by O’Grady. As a first application, we construct a birational model of \tilde{M} which is a compactification of the twisted intermediate Jacobian of the cubic fourfold. Secondly, we show that \tilde{M} is the MRC quotient of the main component of the Hilbert scheme of elliptic quintic curves in the cubic fourfold, as conjectured by Castravet. This is joint work with Chunyi Li and Xiaolei Zhao.

Zoom link: https://ed-ac-uk.zoom.us/j/83313857579

Meeting ID: 833 1385 7579

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In characteristic p, Kollár showed that Fano varieties that are p-cyclic covers can admit differential forms. This is a powerful tool for studying these varieties. By specializing to characteristic p one can use the positivity of these forms to show that the birational geometry complex Fano hypersurfaces can have quite different behavior from the birational geometry of projective space. For example, Kollár used these forms to prove that there are Fano hypersurfaces which are not rational (or even ruled), and Totaro showed that these Fano hypersurfaces are not even stably rational. In this talk we give new applications of these degeneration techniques. We explain that Fano hypersurfaces can have arbitrarily large degrees of irrationality and we show that the degrees of rational endomorphisms of Fano hypersurfaces must satisfy certain congruence constraints modulo p. This is joint work with Nathan Chen.

(Zoom details: see above)

I will report on recent joint work with D. Agostini and K.-W. Lai, where study how the degrees of irrationality of the moduli space of polarized K3 surfaces grow with respect to the genus g. We prove that, for a series of infinitely many genera, the irrationality is bounded by the Fourier coefficients of certain modular forms of weight 11, and thus grow at most polynomially, in terms of g. Our proof relies on results of Borcherds on Heegner divisors together with results of Hassett, Debarre—Iliev—Manivel, and Debarre-Macrì on special hyperkähler fourfolds.

Quot-schemes of locally free quotients of a given sheaf $\mathsrc{G}$ introduced by Grothendieck are generalisations of projectivizations and Grassmannian bundles, and are closely related to degeneracy loci of maps between vector bundles. In this talk we will study the derived categories of these Quot-schemes when the sheaf $\mathsrc{G}$ has homological dimension $\le 1$. This framework allows us to remove the smoothness conditions for many known formulae such as the formulae for blowups, Cayley's trick, standard flips, projectivizations, and Grassmannain type flips, as well as provide new phenomenon such as virtual flips, and formulae for blowups of determinantal ideals, etc. We will illustrate our idea of proof in concrete examples, and if time allowed, we will also discuss some applications to the case of moduli of linear series on curves, and Brill--Noether theory for moduli of stable objects in K3 categories.

Supersingular twistor spaces are certain families of K3 surfaces over A^1 associated to a supersingular K3 surface. We will describe a geometric construction that produces families of K3 surfaces over P^1 which compactify supersingular twistor spaces. We will give some results describing the structure of these families, and in particular the relationship between the fibers over 0 and infinity. Finally, we will give some applications.

In a celebrated series of papers, Mukai established structure theorems for polarized K3 surfaces of all genera g<21, with the exception of the case g=14. Using the identification between certain moduli spaces of polarized K3 surfaces and the moduli space of special cubic fourfolds of given discriminant, we discuss a novel approach to moduli spaces of K3 surfaces. As an application, we establish the rationality of the universal K3 surface of these genus 14 and 22. This is joint work with A. Verra.

When trying to apply the machinery of derived categories in an arithmetic setting, a natural question is the following: for a smooth projective variety X, to what extent can Db(X) be used as an invariant to answer rationality questions? In particular, what properties of Db(X) are implied by X being rational, stably rational, or having a rational point? On the other hand, is there a property of Db(X) that implies that X is rational, stably rational, or has a rational point? In this talk, we will examine a family of arithmetic toric varieties for which a member is rational if and only if its bounded derived category of coherent sheaves admits a full etale exceptional collection. Additionally, we will discuss the behavior of the derived category under twisting by a torsor, which is joint work with Mattew Ballard, Alexander Duncan, and Patrick McFaddin.

The rational Chow ring of the moduli space M_g of curves of genus g is known for g \leq 6. In each of these cases, the Chow ring is tautological (generated by certain natural classes known as kappa classes). In recent joint work with Sam Canning, we prove that the rational Chow ring of M_g is tautological for g = 7, 8, 9, thereby determining the Chow rings by work of Faber. In this talk, I will give an overview of our approach, with particular focus on the locus of tetragonal curves (special curves admitting a degree 4 map to P^1).

If X is a smooth and projective variety, then deformations of X induce deformations of its derived category D(X). It may however happen that D(X) has more deformations than those coming from X. These are sometimes considered 'non-commutative' deformations of X.

Over the dual numbers k[\epsilon], non-commutative deformations have been constructed and classified by Toda. Over more general Artinian rings, such deformations are harder to study because the deformation theory of a derived category (or better, dg category) typically does not satisfy the Schlessinger axioms.

We show that for many dg categories D of Calabi-Yau flavour, the functor parametrising deformations of D is well-behaved and in fact representable by a power series ring. This includes in particular the case D=D(X), for X a variety with trivial canonical bundle. This can be seen as a Bogomolov-Tian-Todorov theorem for non-commutative deformations. We also construct a period map and show a `local Torelli' theorem for non-commutative deformations of hyperkahler varieties and abelian varieties.

This talk is part of the attempt to lift various structures from the cohomology rings of hyperkahler manifolds to their Chow rings. In the situation at hand, we consider the case of Hilb(K3), the Hilbert scheme of points on a K3 surface. We take Oberdieck's lift to Chow of the Looijenga-Lunts-Verbitsky Lie algebra action on H^*(Hilb(K3)), and show that it induces a multiplicative decomposition of the motive of Hilb(K3). Joint work with Georg Oberdieck and Qizheng Yin.

Nodal quintic del Pezzo threefolds (Fano 3-folds of index 2 and degree 5) X_m can have m=1,2,3 nodes and they are unique in each case. Geometrically, X_m are the contraction of a smooth section E=P^1 of a quadric surface fibration Y_m over P^1 to a nodal point. A mutation of Kuznetsov’s semiorthogonal decomposition (SOD) of D^b(Y_m) descends to a Kawamata type decomposition of D^b(X_m). This is a different approach to the same result by Pavic and Shinder. The key points are: (1) the non-trivial component of Kuznetsov’s SOD is equivalent to the derived category of a curve C_m, a chain of m P^1s; (2) the push-forward of the Fourier-Mukai kernel of the embedding D^b(C_m)->D^b(Y_m) to Y_m is the rank 2 spinor bundle of Y_m associated with the smooth section E.

The derived category of a K3 surface is governed by its integral cohomology together with its Hodge structure and to objects we can functorially assign a Mukai vector in this lattice. We show an analogous picture for higher-dimensional hyper-Kähler manifolds using the extended Mukai lattice. In particular, we construct a vector for certain objects in the derived category taking values in the extended Mukai lattice and we obtain a rank 25 integral lattice with a Hodge structure which is a derived invariant for hyper-Kähler manifolds deformation-equivalent to the Hilbert scheme of n points on a K3 surface.

The D-equivalence conjecture, due to Bondal, Orlov, and Kawamata, predicts that a birational equivalence between smooth projective varieties that preserves the canonical bundle should induce an equivalence of derived categories of coherent sheaves. I will give an overview of "window categories" in equivariant derived categories of coherent sheaves, which can be used to construct derived equivalences for birational transformations coming from variation of GIT quotient. I will then discuss how these were used recently to prove the D-equivalence conjecture for projective Calabi-Yau manifolds in the birational equivalence class of a moduli space of sheaves on a K3 surface.

Motivated by classical enumerative geometry and mathematical physics, counting curves in Calabi-Yau 3-folds has been studied intensively for decades, including Gromov-Witten theory and Donaldson-Thomas theory. In recent years, mathematical theory (by Hosono-Saito-Takahashi, Kiem-Li, Maulik-Toda etc) has been developed to realize the idea of Gopakumar and Vafa to recover the curve-counting invariants using the geometry of 1-dimensional sheaves. These developments shed new light on both enumerative geometry and the classical geometry of the relevant moduli spaces. I will discuss 3 particular cases (1) Higgs bundles (2) K3 surfaces, and (3) CP^2, where the Gopakumar-Vafa theory interacts with some other structures and conjectures in a surprising way.

In this talk we address several open questions and generalize the existing results about the uniqueness of enhancements for triangulated categories which arise as derived categories of abelian categories or from geometric contexts. If time permits, we will also discuss applications to the description of exact equivalences. This is joint work with A. Canonaco and A. Neeman.

Punctured surfaces are the simplest class of symplectic manifolds and there are many constructions of homological mirrors for them, i.e. constructions in algebraic geometry of a category equivalent to the Fukaya category of the surface. To make the Fukaya category Z-graded, not just Z/2-graded, you need to choose a line-field on the surface. I'll explain what this choice corresponds to in (some of) the mirror constructions, it leads to a kind of twisted derived category which doesn't seem to have been widely studied.

A triangulated category is said to be indecomposable if it admits no nontrivial semiorthogonal decompositions. For a derived category of coherent sheaves on a variety Y, we propose a stronger condition, which implies, among other things, that for any variety X, any semiorthogonal decomposition of the product X x Y is induced from a decomposition of X. For X = {pt} this implies the usual indecomposability. We show that varieties with finite Albanese morphism, e.g., curves of positive genus, are stably semiorthogonally indecomposable in this sense. From this, we deduce the non-existence of phantom subcategories in the product surfaces C x P^1, where C is a smooth projective curve of positive genus, and in some other examples as well.

Some years ago I constructed a new autoequivalence of the derived category of the Hilbert scheme of n points on a K3 surface using "P-functors." Later Donovan, Meachan, and I extended the construction to some moduli spaces of torsion sheaves, and illuminated the geometric meaning of the story. Now my student Andrew Wray and I can extend it to moduli spaces of sheaves of any rank, powered by a new proof of the standard results about those moduli spaces. We deform to a Hilbert scheme in one step, using twistor lines.

In 2013, as an analogue of Franchetta's classical conjecture on the Picard group of the universal genus g curve, O'Grady asked whether the Chow group of zero-cycles of the generic fiber of the universal genus-g K3 surface is cyclic. I will discuss some recent progress that I obtained in collaboration with Laterveer and Vial on this conjecture as well as its higher dimensional version for hyper-Kähler varieties. The main feature of our argument is the combination of the projective geometry of cubic fourfolds on one hand and moduli spaces of Bridgeland stable objects in their Kuznetsov components on the other hand.

In this talk we consider the (reduced) Donaldson-Thomas theory of the product of a K3 surface and an elliptic curve. In rank 1, these invariants can be viewed as enumerating algebraic curves and are known for fiber classes over the elliptic curve (work of Pandharipande and Thomas), and for classes primitive over the K3 (work of Pixton, Shen and myself). I will explain how to extend these results to arbitrary curve classes. If time permits, I will also give an outlook on the higher rank case (work in progress).

Fix a Calabi-Yau 3-fold X satisfying the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda, such as the quintic 3-fold. I will explain joint works in progress with Richard Thomas that aim to express Joyce’s generalised DT invariants counting Gieseker semistable sheaves of any rank r on X in terms of those counting sheaves of rank 1. By the MNOP conjecture, the latter are determined by the Gromov-Witten invariants of X. Our technique is to use wall-crossing with respect to weak Bridgeland stability conditions on X.

The Serre functor of a triangulated category is one of its most important invariants, playing the role of the dualizing complex of a variety in noncommutative algebraic geometry. I will explain how to describe the Serre functors of many semiorthogonal components of varieties in terms of spherical twists, with applications to a dimension formula for Kuznetsov components of complete intersections conjectured by Katzarkov and Kontsevich, to the nonexistence of Serre invariant stability conditions, and to the construction of Calabi-Yau categories as crepant contractions. This is joint work with Alexander Kuznetsov.

I will talk about a new finiteness theorem for variations of Hodge structure. It is a generalization of the Cattani-Deligne-Kaplan theorem from Hodge classes to so-called self-dual (and anti-self-dual) classes. For example, among integral cohomology classes of degree 4, those of type (4,0) + (2,2) + (0,4) are self-dual, and those of type (3,1) + (1,3) are anti-self-dual. The result is suggested by considerations in theoretical physics, and the proof uses o-minimality and the definability of period mappings. This is joint work with Benjamin Bakker, Thomas Grimm, and Jacob Tsimerman.

Let C be a smooth projective curve of genus at least 2 and let N be the moduli space of stable rank-two bundles on C of odd degree. We construct a semi-orthogonal decomposition in the derived category of N conjectured by Narasimhan and by Belmans, Galkin and Mukhopadhyay. It contains two copies of the i-th symmetric power of C for i=0,...,g-2, one copy of the (g-1)-st symmetric power, and possibly a semi-orthogonal complement to all those blocks. This complement is expected to be trivial by the BGMN conjecture. Our approach is based on an analysis of wall-crossing between moduli spaces of stable pairs, combining classical vector bundles techniques with the method of windows. This is joint work with Jenia Tevelev.

We construct quantum K-invariants in non-archimedean analytic geometry. Our approach differs from the classical one in algebraic geometry via perfect obstruction theory. Instead, we build on our previous works on the foundation of derived non-archimedean geometry, the representability theorem and Gromov compactness. We obtain a list of natural geometric relations between the stacks of stable maps, directly at the derived level, with respect to elementary operations on graphs, namely, products, cutting edges, forgetting tails and contracting edges. They imply immediately the corresponding properties of K-theoretic invariants. The derived approach produces highly intuitive statements and functorial proofs. The flexibility of our derived approach to quantum K-invariants allows us to impose not only simple incidence conditions for marked points, but also incidence conditions with multiplicities. This leads to a new set of enumerative invariants. Our motivations come from non-archimedean enumerative geometry and mirror symmetry. Joint work with M Porta.

Fix a Calabi-Yau 3-fold X. Its DT invariants count stable bundles and sheaves on X. The generalised DT invariants of Joyce-Song count semistable bundles and sheaves on X. I will describe work with Soheyla Feyzbakhsh using the weak stability conditions of Bayer-Macrì -Toda to show these generalised DT invariants in any rank r can be written in terms of rank 1 invariants. By the MNOP conjecture the latter are determined by the GW invariants of X.

The isomorphism between the first Hirzebruch surface and the blowup of the projective plane in a point is a well-known result, due to Hirzebruch. From a numerical classification of Grothendieck groups of rank 4 which behave like the Grothendieck group of a smooth projective surface we expect the existence of exotic noncommutative surfaces, which are surfaces not obtained via deforming commutative surfaces. There exist two constructions: one as an asymmetric noncommutative P^1-bundle (due to de Thanhoffer--Presotto), one as a fat point blowup (joint work with Presotto). I will explain how a Hirzebruch isomorphism for these two families of surfaces exists as an equivalence of (derived) categories, and how this is related to some very classical geometry of linear systems. This is joint work with Dennis Presotto and Michel Van den Bergh.

We will define residual categories of Lefschetz exceptional collections and discuss some conjectural relations between the structure of quantum cohomology and residual categories. We will illustrate this relationship in the case of some isotropic Grassmannians. Based on joint works with Alexander Kuznetsov.

Quotient varieties for reductive groups admit the Kirwan (partial) resolution of singularities, and quite often also a noncommutative crepant resolution (NCCR). We will construct a (Orlov's type) semi-orthogonal decomposition of the derived category of the Kirwan resolution which has the derived category of the NCCR as a component, and where other components also have a concrete (almost) geometric description. This is a joint work with Michel Van den Bergh.

A compact hyper-Kähler manifold is a higher dimensional generalization of K3 surfaces. An elliptic fibration of a K3 surface correspondingly generalizes to the so-called Lagrangian fibration of a compact hyper-Kähler manifold. It is known that an elliptic fibration of a K3 surface is always "self-dual" in a certain sense. This turns out to be not the case for higher-dimensional Lagrangian fibrations. In this talk, I will propose a construction for the dual Lagrangian fibration of all currently known examples of compact hyper-Kähler manifolds.