Algebraic Geometry and Moduli



Woonam Lim (UC San Diego)     March 27, 17:00-18:30

Virtual χ-y-genera of Quot schemes on surfaces    

Oprea and Pandharipande studied the virtual Euler characteristic of Quot schemes on surfaces. Based on the calculations in several cases, they conjectured the rationality of the generating series of the virtual invariants. In this talk, I will explain the virtual χ-y-genera of Quot schemes, thus refining the work of [OP]. The main result expresses the Quot scheme invariants universally in terms of the Seiberg-Witten invariants of Durr, Kabanov, and Okonek. Based on calculations in several cases and the blow up formula, we resolve the χ-y-genera analogue of the rationality conjecture of [OP], for all surfaces with pg>0. In addition, the reduced Quot scheme invariants for K3 surfaces and primitive curve classes will be discussed with connections to the Kawai-Yoshioka formula.

Jim Bryan (UBC Vancouver)     April 1, 17:00-18:30

K3 surfaces with symplectic group actions, enumerative geometry, and modular forms    

The Hilbert scheme parameterizing n points on a K3 surface X is a holomorphic symplectic manifold with rich properties. In the 90s it was discovered that the generating function for the Euler characteristics of the Hilbert schemes is related to both modular forms and the enumerative geometry of rational curves on X. We show how this beautiful story generalizes to K3 surfaces with a symplectic action of a group G. Namely, the Euler characteristics of the G-fixed Hilbert schemes parametrizing G-invariant collections of points on X are related to modular forms of level |G| and the enumerative geometry of rational curves on the stack quotient [X/G] . These ideas lead to some beautiful new product formulas for theta functions associated to root lattices. The picture also generalizes to refinements of the Euler characteristic such as χy genus and elliptic genus leading to connections with Jacobi forms and Siegel modular forms.

Drew Johnson (ETHZ)     April 3, 17:00-18:30

Rationality of descendant series for Hilbert and Quot schemes of surfaces    

The Quot scheme of a trivial bundle on a surface with 1-dimensional quotients carries a virtual tangent bundle from which the virtual Euler characteristic can be defined. A recent paper of Oprea and Pandharipande conjectured that the generating series of these numbers are all Laurent expansions of rational functions and checked several cases. In this talk, I will discuss a preprint (joint with Oprea and Pandharipande) which (along with work of W. Lim) completes the proof of the conjecture when the trivial bundle has rank one and introduces a generalization of the conjecture to descendant series. For the descendant series, Chern classes of tautological bundles are inserted along with the tangent bundle. We can prove that the descendant series are rational in the rank one case and the case when the quotients are of dimension 0.

Younghan Bae (ETHZ)     April 8, 17:00-18:30

Relations on the universal Picard stack and some applications    

The universal Picard stack is a stack which parameterizes pointed nodal curves and line bundles. This stack is smooth Artin stack locally of finite type. In this talk we use operational Chow theory approach to define tautological relations on this stack and discuss tautological relations arising from Pixton's formula of double ramification cycles. In the second half, we discuss possible applications of those relations to the Gromov-Witten theory of K3 surfaces. In low genus, we will see how the Katz-Klemm-Vafa formula is related to other descendant invariants. The first half is joint work in progress with D. Holmes, R. Pandharipande, J. Schmitt, R. Schwarz and the second half is joint work in progress with T. Buelles.

Izzet Coskun (UI Chicago)     April 10, 17:00-18:30

The stabilization of the cohomology of moduli spaces of sheaves on surfaces    

The Betti numbers of the Hilbert scheme of points on a smooth, irreducible projective surface have been computed by Goettsche. These numbers stabilize as the number of points tends to infinity. In contrast, the Betti numbers of moduli spaces of semistable sheaves on a surface are not known in general. In joint work with Matthew Woolf, we conjecture these also stabilize and that the stable numbers do not depend on the rank. We verify the conjecture for large classes of surfaces. I will discuss our conjecture and provide the evidence for it.

Angela Gibney (Rutgers)     April 15, 17:00-18:30

Vertex algebras of CohFT-type    

Representations of vertex algebras of CohFT-type can be used to construct vector bundles of coinvariants and conformal blocks on moduli spaces of stable pointed curves. The name comes from the fact that such bundles define semisimple cohomological field theories. I'll present motivation for why one may be interested in such bundles and their classes, as well as a few examples.

Ignacio Barros (NEU Boston)     April 17, 17:00-18:30

On product identities and the Chow rings of holomorphic symplectic varieties    

For a moduli space M of stable sheaves over a K3 surface X, we propose a series of conjectural identities in the Chow rings CH*(M x Xl), l >= 1, generalizing the classic Beauville-Voisin identity for a K3 surface. We emphasize consequences of the conjecture for the structure of the tautological subring R*(M) in CH*(M). We prove the proposed identities when M is the Hilbert scheme of points on a K3 surface. This is joint work with L. Flapan, A. Marian, and R. Silversmith.

Dhruv Ranganathan (Cambridge)     April 22, 17:00-18:30

Gromov-Witten theory and logarithmic intersection theory    

The concrete part of this talk will be a discussion of the degeneration formula for Gromov-Witten theory in SNC degenerations, using the geometry of expanded degenerations. I'll then use this formula to motivate a phenomenon in intersection theory that has recently appeared in the work of a number of people (Herr, Holmes-Pixton-Schmitt, Marcus-Wise). Specifically, there is a certain elementary but infinite dimensional (co)homology theory for algebraic varieties equipped with a SNC divisor, or more generally a logarithmic structure, where natural formulas can be written down. These include fundamental structures like the degeneration formula, localization formula, and product formula for SNC geometries. As I'll try to explain, while this cohomology theory is infinite dimensional, it's only infinite dimensional in a "combinatorial" direction. For example, for toric varieties, the entire ring can be described explicitly, going back to work of Fulton-Sturmfels, Morelli, and others.

Noah Arbesfeld (Imperial College London)     April 24, 17:00-18:30

Box counting and Quot schemes    

I'll present concise expressions for the generating series of equivariant virtual holomorphic Euler characteristics of the Quot schemes parametrizing 0-dimensional quotients on C2 and C3. These expressions can also be regarded as combinatorial identities involving 2- and 3-dimensional partitions, respectively. The formulas for quotients on C2 yield information about certain so-called Nekrasov partition functions, Euler characteristics of sheaves on instanton moduli space. The formulas for threefolds provide some evidence for a conjectural framework of Nekrasov and Okounkov relating the Donaldson-Thomas theory of threefolds with the geometry of certain Calabi-Yau fivefolds. This is joint work with Yakov Kononov.

Sheldon Katz (UI Urbana-Champaign)     April 29, 17:00-18:30

Generalized Kac-Moody Lie algebras and moduli spaces of 1-dimensional sheaves on surfaces

Consider a smooth projective surface S and a positive integer d. For each curve class beta on S we have a moduli space Mβ of stable sheaves on S of dimension 1, class β, and euler characteristic 1. Let Vd be the direct sum of the H*(Mβ) over all β with KS . β = - d. Then a representation of SL(2) on Vd can be associated to any -2 curve on S. If we have several -2 curves whose intersection matrix is the negative of a generalized Cartan matrix, the associated SL(2) representations generate a representation of the associated GKM algebra. If the GKM is affine, then d is identified with the level of this representation. These Lie algebra actions all commute with the SL(2)xSL(2) representation constructed by Kiem and Li in the context of Gopakumar-Vafa invariants. As an example, we arrive at a mathematical explanation of the affine E8 global symmetry of the refined E-string of physics. This talk is based on joint work with Davesh Maulik.

Yalong Cao (IPMU Tokyo)     May 1, 16:00-17:30

Curve counting via stable objects in derived categories of Calabi-Yau 4-folds    

In a joint work with Davesh Maulik and Yukinobu Toda, we proposed a conjectural Gopakumar-Vafa type formula for the generating series of stable pair invariants on Calabi-Yau 4-folds. In this talk, I will present the recent joint work with Yukinobu Toda on how to give an interpretation of the above GV type formula in terms of wall-crossing phenomena in the derived category of coherent sheaves.

Johannes Schmitt (Bonn)     May 6, 17:00-18:30

Strata of k-differentials and double ramification cycles    

Inside the moduli space of stable curves we have the strata of k-differentials, the closures of loci of smooth pointed curves admitting a meromorphic k-differential form with zeros and poles of specified orders at the marked points. A conjecture relating the fundamental classes of these strata to Pixton's formula for the (generalized) double ramification cycle was proposed by Janda, Pandharipande, Pixton, Zvonkine (for k=1) and myself (for k>1). I will recall the conjecture and explain how it can now be proved using recent joint work with Bae, Holmes, Pandharipande and Schwarz on double ramification cycles in the universal Picard stack.

Paul Ziegler (TU Munich)     May 8, 17:00-18:30

Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration    

I will talk about a proof, joint with M. Gröchenig and D. Wyss, of the topological mirror symmetry conjecture of Hausel and Thaddeus, which predicts the equality of suitably defined Hodge numbers of moduli spaces of Higgs bundles with SL(n)- and PGL(n)-structure. The proof, inspired by an argument of Batyrev, proceeds by comparing the number of points of these moduli spaces over finite fields via p-adic integration.

Maria Yakerson (Regensburg)     May 13, 17:00-18:30

The Hilbert scheme of infinite affine space    

Various invariants have been computed for Hilbert schemes of surfaces, however our knowledge about Hilbert schemes (of points) of higher dimensional schemes is quite limited. For example, Hilbert schemes of n-dimensional affine spaces have very complicated geometry for high n. In this talk we will present the surprising observation, that the Hilbert scheme of infinite dimensional affine space has homotopy type of a Grassmannian, and so its invariants of homotopical nature have a simple description. We will explain then how this observation allows us to obtain new properties of algebraic and hermitian K-theories as generalized cohomology theories. This is joint work with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, and Burt Totaro.

Oliver Leigh (Stockholm)     May 15, 17:00-18:30

The r-ELSV formula via the moduli space of stable maps with divisible ramification    

For a fixed positive integer r, a stable map is said to have "divisible ramification" if the order of every ramification locus is divisible by r. In this talk I'll discuss a theory of stable maps with divisible ramification and show how this leads to a new geometric framework from which to view and prove Zvonkine's r-ELSV formula. Namely, that there is a natural moduli space which parametrises these objects and how the construction of this space leads to a natural Hurwitz-type theory where the techniques of virtual localisation and degeneration of target can be applied.

Emanuele Macrì (Orsay)     May 20, 17:00-18:30

Antisymplectic involutions on projective hyperkähler manifolds    

An involution of a projective hyperkähler manifold is called antisymplectic if it acts as (-1) on the space of global holomorphic 2-forms. I will present joint work in progress with Laure Flapan, Kieran O'Grady, and Giulia Saccà on antisymplectic involutions associated to polarizations of degree 2. We study the number of connected components of the fixed loci and their geometry.

Michel van Garrel (Warwick)     May 22, 17:00-18:30

Mixing curve counting theories and geometries    

I will describe a series of unexpected formulas that relate counts of curves in log CY surfaces, counts of disks in toric CY threefolds, counts of D-branes in local CY fourfolds and representations of quivers. At the end, I will elaborate on a prediction for refined BPS invariants of local CY fourfolds. Joint work with Pierrick Bousseau and Andrea Brini.

Felix Janda (IAS Princeton) and Dimitri Zvonkine (CNRS/Versailles)     May 27, 17:00-19:00

The locus of curves with holomorphic differentials and Witten's r-spin class    

In the moduli space of stable pointed curves we define the holomorphic locus: it is the locus of curves that carry holomorphic 1-forms with zeros of prescribed orders at the marked points. Our goal is to express the colomology class Poincaré dual to this locus. We present an approach to this question via Witten's r-spin class. For each value of r, Witten's classes form a cohomological field theory, which allows one to write an explicit expression for these classes. It turns out that the dependence of this expression on r is polynomial for large r. We conjecture that the value of the polynomial at r=0 is the cohomology class of the holomorphic locus (up to an explicit sign). We present the plan of a proof of this conjecture based on equivariant localization and the cosection construction of virtual fundamental classes. This is work in progress with Qile Chen, Yongbin Ruan, and Adrien Sauvaget.

Aaron Pixton (Michigan)     May 29, 17:00-18:30

Boundary vanishing, kappa rings, socle evaluations, and λg formulas    

I will describe work in progress dealing with a sequence of interesting tautological classes on the moduli space of stable curves with connections (some conjectural) to the following topics: the kappa ring of the moduli space of curves of compact type, socle evaluation maps on tautological rings, and the double ramification cycle formula for λg.

Pierrick Bousseau (ETHZ)     June 3, 17:00-18:30

Quasimodular forms from Betti numbers    

This talk will be about refined curve counting on local P2, the noncompact Calabi-Yau 3-fold total space of the canonical line bundle of the projective plane. I will explain how to construct quasimodular forms starting from Betti numbers of moduli spaces of one-dimensional coherent sheaves on P2. This gives a proof of some stringy predictions about the refined topological string theory of local P2 in the Nekrasov-Shatashvili limit. This work is in part joint with Honglu Fan, Shuai Guo, and Longting Wu.

Alexander Kuznetsov (Steklov Institute, Moscow)     June 5, 17:00-18:30

Moduli spaces of Gushel-Mukai varieties    

Gushel-Mukai varieties are Fano varieties with Picard number 1, coindex 3, and degree 10; their dimension ranges between 3 and 6. These varieties have very interesting geometry. In the talk, I will review some basic facts about these varieties and discuss their moduli stacks and moduli spaces, which, surprisingly, have a very explicit description.

Gavril Farkas (HU Berlin)     June 12, 17:00-18:30

The Kodaira dimension of the moduli space of curves of genus 22 and 23    

We prove that the moduli spaces of curves of genus 22 and 23 are of general type. To do this, we calculate certain virtual divisor classes of small slope associated to certain linear series with special quadric relations. We then develop new tropical methods for studying linear series and independence of quadrics and show that these virtual classes are represented by effective divisors. This amounts to proving the Strong Maximal Rank Conjecture for these cases. Joint work with Jensen and Payne.

Eric Riedl (Notre Dame)     June 17, 17:00-18:30

Linear sections of hypersurfaces    

Given a hypersurface X in Pn, we can consider the family of all linear sections of X. One would generally expect the linear sections to vary as much as possible, but singular hypersurfaces with automorphisms provide counterexamples. We relate this question to results on the semi-stability of the log tangent sheaf restricted to a general linear section. We use this to completely classify the curves in P2 that do not have maximal variation of linear sections. Using a strengthened version of Grauert-Mulich, we also prove maximal variation of hyperplane sections for smooth hypersurfaces of general type with d > n+2. This is joint work with Dennis Tseng and Anand Patel.

Martin Möller (Frankfurt)     June 19, 17:00-18:30

The moduli space of multi-scale differentials    

The moduli space of multi-scale differentials is a smooth DM-stack that compactifies the moduli space of abelian differentials of fixed type of zeros. We present the functor of multi-scale differentials with consequences for the local structure near the boundary. Applications to the structure of the tautological ring and to the Chern classes of this moduli space are given. Joint with Bainbridge-Chen-Gendron-Grushevsky and with Costantini-Zachhuber.

Lothar Göttsche (ICTP Trieste)     June 24, 17:00-18:30

Lehn and Verlinde formulas for moduli of sheaves on surfaces    

This is a report on joint work with Martijn Kool. Recently, Marian-Oprea-Pandharipande established (a generalization of) Lehn's conjecture for Segre numbers associated to Hilbert schemes of points on surfaces. Extending work of Johnson, they provided a conjectural correspondence between Segre and Verlinde numbers. For surfaces with holomorphic 2-form, we propose conjectural generalizations of their results to moduli spaces of stable sheaves of higher rank. Using Mochizuki's formula, we derive a universal function which expresses virtual Segre and Verlinde numbers of surfaces with holomorphic 2-form in terms of Seiberg- Witten invariants and intersection numbers on products of Hilbert schemes of points. We use this to verify our conjectures in examples.

Miguel Moreira (ETHZ)     June 26, 17:00-18:30

Virasoro constraints for stable pairs on 3-folds and for Hilbert schemes of points of surfaces    

The theory of stable pairs (PT) with descendents, defined on a 3-fold X, is a sheaf theoretical curve counting theory. Conjecturally, it is equivalent to the Gromov-Witten (GW) theory of X via a universal (but intricate) transformation, so we can expect that the Virasoro conjecture on the GW side should have a parallel in the PT world. In joint work with A. Oblomkov, A. Okounkov and R. Pandharipande, we formulated such a conjecture and proved it for toric 3-folds in the stationary case. The Hilbert scheme of points on a surface S might be regarded as a component of the moduli space of stable pairs on S x P1, and the Virasoro conjecture predicts a new set of relations satisfied by tautological classes on S[n]. I will explain how to prove this result for an arbitrary simply-connected S by reducing it to the toric case via the existence of universal formulas for integrals of tautological classes on S[n].

Burt Totaro (UCLA)     July 1, 17:00-18:30

The Hilbert scheme of n-dimensional affine space    

I will recall our description of the homotopy type of the Hilbert scheme of points on affine space An in the limit where n goes to infinity. The talk will prove sharper results on the topology of Hilbd(An) for finite n. We also formulate a conjecture on the Hilbert scheme of X x A for any smooth variety X. Joint with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, and Maria Yakerson.

Mark Gross (Cambridge)     July 3, 17:00-18:30

The higher dimensional tropical vertex and the canonical scattering diagram

I will talk about two pieces of work, one joint with Arguz, one joint with Siebert. This work generalizes to higher dimension the tropical vertex of Gross-Pandharipande-Siebert in the first instance, and the canonical scattering diagram of Gross-Hacking-Keel in the second instance. The canonical scattering diagram of a log Calabi-Yau pair (X,D) is a data structure which records counts of rational curves meeting D at one point. The tropical vertex produces scattering diagrams via an algorithm whose two-dimensional version was introduced by Kontsevich and Soibelman. The main result shows how to compare the two scattering diagrams in the case that (X,D) is obtained by blowing up a toric variety.

Jørgen Rennemo (Oslo)     July 8, 17:00-18:30

Fixed-point loci in the Hilbert schemes of points in the plane    

The Betti numbers of Hilbert schemes of points in the plane were first computed by Ellingsrud and StrÝmme. In this talk, we'll let a finite cyclic group act on the plane, and consider the analogous problem of computing the Betti numbers of the fixed loci under the induced action on the Hilbert scheme of points. The main result is a new product formula for the Betti numbers in the case where no element of the group fixes the symplectic form on the plane. This is joint work with Paul Johnson.

Arend Bayer (Edinburgh)     July 10, 17:00-18:30

Brill-Noether in K3 categories and applications

I will present a general conjecture on Brill-Noether type statements for moduli spaces of stable objects in K3 categories. I will explain cases that hold in general, as well as specific situations where the claim holds for more elementary geometric reasons. In applications I will focus on cases of conjecturally rational cubic fourfolds. This is based on joint work with Aaron Bertram, Emanuele Macri and Alex Perry, and with Huachen Chen and Qingyuang Jiang.

Dan Abramovich (Brown)     July 15, 17:00-18:30

Resolution and logarithmic resolution via weighted blowings up    

This lecture combines resolution of singularities, logarithmic geometry and algebraic stacks. I will not assume familiarity neither with resolution of singularities nor with logarithmic geometry. I report on work with Temkin and Wlodarczyk and work of Quek. Resolving singularities in families requires logarithmic geometry. Surprisingly, trying to do this canonically forces us to use stack-theoretic modifications. Surprisingly, stack-theoretic modifications provides an efficient iterative resolution method in which the worst singularities are blown up without regard to the history. Not so surprisingly, to make exceptional divisors cooperate we need logarithmic geometry again.

Dragos Oprea (UC San Diego)     July 22, 17:00-18:30

The virtual K-theory of Quot schemes of surfaces    

The Quot schemes of surfaces parametrizing quotients of dimension at most 1 of the trivial sheaf carry 2-term perfect obstruction theories. We prove that the generating series of virtual K-theoretic invariants are given by rational functions for several geometries. Given a K-theory class of rank r on the surface, we associate natural series of virtual Segre and Verlinde numbers. For punctual quotients of the trivial sheaf of rank N, we prove that the Segre and Verlinde series match, and establish a new symmetry exchanging r and N. The Segre/Verlinde statements have analogues over curves. This is based on joint work with Noah Arbesfeld, Drew Johnson, Woonam Lim and Rahul Pandharipande.