Algebraic Geometry Seminar

Let $C$ be a compact Riemann surface, and $\Gamma=\pi_1(C)$ its fundamental group. While enumerating the representations of $\Gamma$ over finite fields, Hausel and Rodrigues-Villegas have noticed that they always obtained palindromic polynomials. One way to interpret this symmetry is to say that the cohomology of the variety $\mathcal{M}_B$ of representations of $\Gamma$ over complex numbers (so called character variety) admits a "curious" Poincaré duality. Similar dualities were previously studied by de Cataldo and Migliorini, but their theory could only be applied to the homeomorphic variety $\mathcal{M}_D$ of Higgs bundles on $C$. Matching up these two dualities (the "curious" one being conjectural) boils down to an equality of two filtrations (perverse and weight) on $H^*(\mathcal{M}_B)$ of very different natures. This equality became known as $P=W$ conjecture. In my talk, I will explain this story in more detail, and give some indication of how this conjecture was eventually proved. Based on joint work with T. Hausel, A. Mellit, O. Schiffmann.

The main technical tool in our proof of $P=W$ conjecture was the construction of an action of a certain algebra, which we called deformed $W_{1+\infty}$-algebra, on the cohomology of the moduli space of stable Higgs bundles on the curve $C$. It turns out that this algebra is nothing else than the cohomological Hall algebra of finite length sheaves on $T^*C$ (see my talk at Academia Sinica in 2019). This isomorphism can be extended to any (cohomologically pure) surface $S$, which opens the way for algebraic study of cohomology of various moduli spaces of sheaves. I will explain the proof of this isomorphism, and speculate about applications to $P=C$ phenomena for $K3$ surfaces and computation of ring structure of $H^*(Hilb S)$ and its cousins. Based on joint work with A. Mellit, O. Schiffmann, E. Vasserot.

We propose a theory of enumerative invariants for structure groups of type B/C/D, that is, for the orthogonal and symplectic groups. For example, we count orthogonal or symplectic principal bundles on projective varieties, or a quiver analogue called self-dual quiver representations. We also discuss algebraic structures arising from the relevant moduli spaces. In type A, Joyce constructed a somewhat mysterious vertex algebra structure on the homology of moduli spaces, and in type B/C/D, we construct a twisted module for this vertex algebra. We then use these algebraic structures to write down wall-crossing formulae for our invariants.

Following our reformulation (joint with Lim-Moreira) of sheaf-theoretic Virasoro constraints with applications to curves and surfaces, I will talk about the quiver analog. After phrasing a universal approach to Virasoro constraints for moduli of quiver-representations, I will sketch their proof for any finite quiver with relations, with frozen vertices, but without cycles. I will use partial flag varieties as a guiding example throughout, following our reformulation (joint with Lim-Moreira) of sheaf-theoretic Virasoro constraints with applications to curves and surfaces, I will talk about the quiver analog. After phrasing a universal approach to Virasoro constraints for moduli of quiver-representations, I will sketch their proof for any finite quiver with relations, with frozen vertices, but without cycles. I will use partial flag varieties as a guiding example throughout but the most exciting upshot is an independent, self-contained proof of Virasoro constraints for Mumford (semi)stable torsion-free sheaves on the complex projective plane but the most exciting upshot is an independent, self-contained proof of Virasoro constraints for Mumford (semi)stable torsion-free sheaves on the complex projective plane.

The Atiyah-Bott localization theorem says that the equivariant cohomology of a space can be recovered, up to inverting some elements, from the equivariant cohomology of the fixed point subspace. We discuss a categorified version of this result which allows us to deduce the theorem for all oriented theories (cohomology and Borel-Moore homology). This is based on a joint work with Adeel Khan.

The existence of Kaehler-Einstein metrics on Fano 3-folds can be determined by studying some positive numbers called stability thresholds. K-stability is ensured if appropriate bounds can be found for these thresholds. An effective way to verify such bounds is to construct flags of point-curve-surface inside the Fano 3-folds. This approach was initiated by Abban-Zhuang, and allows us to restrict the computation of bounds for stability thresholds only on flags. We employ this machinery to prove K-stability of terminal quasi-smooth Fano 3-fold hypersurfaces. Many of these varieties had been attacked by Kim-Okada-Won using log canonical thresholds. In this talk I will tackle the remaining Fano hypersurfaces via Abban-Zhuang Theory.

For complex manifolds, the Riemann-Hilbert correspondence generalizes the classical correspondence between finite dimensional local systems and vector bundles with connections to perverse sheaves and regular holonomic D-modules. These later objects are in fact microlocal in nature that they can be regarded as living on the cotangent bundles, and the correspondence admits a microlocalization as well. Continuing from an earlier joint work with Cote, Nadler, and Shende, we will globalize this correspondence to general complex contact manifolds in an upcoming work.

Given a smooth algebraic surface S over the complex numbers, the Hilbert scheme of points of S is the starting point for many investigations, leading in particular to generating functions with modular behaviour and Heisenberg algebra representations. I will explain aspects of a similar story for surfaces with rational double points, with links to Nakajima quiver varieties, algebraic combinatorics and the representation theory of simple and affine Lie algebras. My talk will be based on a series of joint papers with Bertsch, Craw, Gammelgaard, Gyenge and Nemethi.

Quantum intersection numbers were introduced through a natural quantization of the KdV hierarchy in a work of Buryak, Dubrovin, Guere, and Rossi. Because of the Kontsevich-Witten theorem, a part of the quantum intersection numbers coincides with the classical intersection numbers of psi-classes on the moduli spaces of stable algebraic curves. I will talk about our joint work in progress with Xavier Blot, where we relate the quantum intersection numbers to the stationary relative Gromov-Witten invariants of the Riemann sphere, with an insertion of a Hodge class. Using the Okounkov-Pandharipande approach to such invariants (with the trivial Hodge class) through the infinite wedge formalism, we then give a short proof of an explicit formula for the ``purely quantum'' part of the quantum intersection numbers, found before by Xavier, which in particular relates these numbers to the one-part double Hurwitz numbers.

The anticanonical system is probably one of the most natural objects associated with a Fano manifold. In this talk I will present some examples of Fano manifolds with nonempty anticanonical base locus and discuss a new result on the anticanonical system of Fano fourfolds: if the base locus is a normal irreducible surface, then all of its members are singular. Joint work with Andreas Höring.

The canonical 4-term sequence on moduli spaces of quiver representations is an important tool to study their geometry. First I will use it to describe the Chow ring in more detail. Next, I will use it to prove the rigidity of quiver moduli, and describe their infinitesimal symmetries. Finally, I will use it to establish Schofield's partial tilting conjecture, and obtain a fully faithful Fourier-Mukai functor. The stacky perspective of these moduli spaces, and Teleman quantization, plays an important role. This surveys several joint works with Ana-Maria Brecan, Hans Franzen, Gianni Petrella, and Markus Reineke

Stability conditions are a framework to study moduli of complexes. In fact, the collection of all stability conditions forms a complex manifold called the stability manifold. Understanding the topology and geometry of the stability manifold has applications to homological mirror symmetry, representation theory, symplectic geometry, and the moduli of stable sheaves. In this direction, a folklore conjecture states the stability manifold is actually contractible. In this talk we give a partial answer to this conjecture in the case of surfaces.

Gromov-Witten invariants give a virtual count of the number of curves on a smooth projective variety with given conditions. In general, Gromov-Witten invariants are rational numbers due to multiple cover contributions. To isolate contributions not involving multiple covers, people define Gopakumar-Vafa type invariants (particularly on certain projective varieties) and conjecture their integrality.

In this talk, I will review the genus zero multiple cover formula on semi-positive varieties and define the genus zero Gopakumar--Vafa type invariants. Finally, I will outline the proof of the integrality of Gopakumar--Vafa type invariants in this case. The main technique is to relate Gopakumar--Vafa type invariants to quantum $K$-invariants and to utilize the integrality of the latter.

Bielliptic surfaces form a particular class in the Enriques-Kodaira classification of compact minimal complex surfaces. They possess an explicit description as the quotient of a product of elliptic curves by a group action. They were classified in seven subfamilies by Bagnera-Franchis at the beginning of the XXth century. In this talk, we will care about the computation of their Gromov-Witten invariants with point insertions and a lambda-class insertion. Using J. Li's decomposition formula, we relate these Gromov-Witten invariants to a graph count with some multiplicities. It is thus possible to perform explicit computations. Using some combinatorial tools, we are able to prove the quasi-modularity of the generating series of the above invariants.

I will outline a logarithmic enhancement of the Gromov-Witten/Donaldson-Thomas correspondence, with descendants, and study the behaviour of the correspondence under simple normal crossings degenerations. I will explain a strong form of the degeneration formula in logarithmic DT (and GW) theory - the numerical DT invariants of the general fiber of a degeneration are determined by the numerical DT invariants attached to strata of the special fiber. As a consequence, we prove compatiblity of the new logarithmic GW/DT correspondence with degenerations, and in particular, that knowledge of the conjecture on the strata of the special fiber of a degeneration implies it on the general fiber. Time permitting, I will try to give a sense for where this recent technical development puts us in terms of knowledge of the descendant GW/DT correspondence in general. The talk is based on recent and ongoing joint work with Davesh Maulik (MIT).

It is known that the cohomology ring of the moduli space of stable Higgs bundles on a curve can be generated by the tautological classes by a result of E. Markman. In this talk, I will explain how to generalize this result to the moduli spaces of stable irregular parabolic Higgs bundles on a curve (with or without fixing the polar parts). The key ingredient is to employ a spectral correspondence due to Kontsevich-Soibelman and Diaconescu-Donagi-Pantev.

On smooth quasi-projective toric 3-folds, vertices are the contributions from an affine toric chart to the enumerative invariants of Donaldson-Thomas (DT) or Pandharipande-Thomas (PT) moduli spaces. Unlike partition functions, vertices are fundamentally torus-equivariant objects, and they carry a great deal of combinatorial complexity, particularly in equivariant K-theory. In joint work with Nick Kuhn and Felix Thimm, we give two different proofs of the K-theoretic DT/PT vertex correspondence. Both proofs use equivariant wall-crossing in a setup originally due to Toda. A crucial new ingredient is the construction of symmetrized pullbacks of symmetric obstruction theories on Artin stacks, using Kiem-Savvas' étale-local notion of almost-perfect obstruction theory.

The topological vertex, developed by Aganagic, Klemm, Marino and Vafa, provides an explicit algorithm to compute the open Gromov-Witten invariants of smooth toric Calabi-Yau threefolds in mathematics, as well as the A-model topological string amplitudes in physics. In this talk, I will introduce our recent work on establishing the connection between the topological vertex and multi-component KP hierarchy. This talk is based on a joint work with Zhiyuan Wang and Jian Zhou.

In this talk, we will review the current study of mirror symmetry for flag varieties. We will focus more on the construction of Landau-Ginzburg model, and discuss a folklore mirror symmetry expectation on the eigenvalues of the first Chern class, using concrete examples of flag varieties of Lie type A. This is based on a work in progress joint with Konstanze Rietsch, Mingzhi Yang and Chi Zhang.

Logarithmic and orbifold structures provide two independent ways to model curves in a variety with tangency along a normal crossings divisor. The associated systems of Gromov-Witten invariants benefit from complementary techniques; this has motivated extensive interest in comparing the two approaches.

I will report on recent work in which we establish a complete comparison which, crucially, incorporates negative tangency orders. Negative tangency orders appear naturally in the boundary splitting formalisms of both theories. As such, our comparison opens the way for the wholesale importation of techniques from one side to the other. Contemporaneous work of Sam Johnston uses our comparison to give a new proof of the associativity of the Gross-Siebert intrinsic mirror ring.

Along the way, I will discuss the pathological geometry of negative tangency mapping spaces, and how this can be described and controlled tropically. A crucial step in our work is the discovery of a "refined virtual class" on the logarithmic moduli space, which gives rise to a distinguished sector of the Gromov-Witten theory.

This is joint work with Luca Battistella and Dhruv Ranganathan.

Hurwitz numbers count covers of Riemann surfaces with fixed ramification data. Via monodromy representation, this can be translated into a count of certain tuples of elements of the symmetric group. This translation, resp. degeneration, allows to express Hurwitz numbers in terms of tropical covers, which offers the opportunity to study structural properties. Recently, variants of Hurwitz numbers have been introduced, which can partially also be expressed in terms of the symemtric group. Among those are twisted Hurwitz numbers, for which we define and study tropical analogues. This is joint work with Marvin Anas Hahn.

Smooth and proper DG-categories are noncommutative versions of smooth and proper schemes. They also include finite dimensional algebras of finite global dimension. Kuznetsov and Shinder defined reflexive DG-categories as a (vast) generalisation; they include all projective schemes and all finite dimensional algebras. Smooth and proper DG-categories can also be characterised as the dualizable objects in the monoidal category of DG-categories localised at Morita equivalences. I’ll explain how by using a monoidal characterisation of reflexive DG-categories, one can show that the Hochschild cohomology of a reflexive DG-category is isomorphic to that of its derived category of cohomologically finite modules.

We present a notion of spin structure for a perfect complex E (on a complex variety X) which is equipped with an oriented quadratic structure. The existence of a spin structure on E is obstructed by a natural Z/2Z-gerbe over X, carrying a universal spin structure. As an application we show how to construct a twisted virtual structure sheaf on moduli spaces of sheaves on Calabi-Yau fourfolds. A-priori, it exists as a Z/2Z-graded sheaf on the spin-gerbe over the moduli space M. We expect that it induces a natural class in the K-theory of M with 2 inverted, which agrees with the Oh-Thomas class.

On Calabi-Yau 4-folds, there are 3 surface counting theories related by change of polynomial Bridgeland stability condition: DT, PT_0, PT_1. For toric Calabi-Yau 4-folds, I discuss the fixed loci of the moduli spaces for these theories with respect to the 3-dimensional torus preserving the Calabi-Yau volume form. When these fixed loci are 0-dimensional and reduced, I present a method to calculate these invariants by virtual localization using vertex, edge, and face terms (and modulo taking square roots). The DT case also appeared in the physics literature in work of Nekrasov-Piazzalunga. This is joint work with Y. Bae and H. Park.

I will report recent progress on the cohomology of the moduli of one-dimensional sheaves on the projective plane. This moduli space relates intimately to the enumerative geometry of local P^2 via a certain perverse filtration on its cohomology, which is in turn determined by the cohomology ring structure. I will explain a systematic approach to study this cohomology ring. Our main ingredients include Mumford-type geometric relations and a representation of the (half) Virasoro algebra on the cohomology of the moduli spaces. Time permitting, I will also discuss a conjectural description of the perverse filtration in terms of explicit tautological classes. Based on joint work with Y. Kononov, W. Lim, and M. Moreira.

I will discuss the mixed-spin-p-fields method for Calabi-Yau quintics. The geometric setup of the method consists of p-fields reformulation of Gromov-Witten theory and FJRW theory, and the usage of master space technique connecting two GIT quotients corresponding to GW and FJRW invariants respectively. This method has been successfully applied to solve BCOV Feynman summation rule for mentioned Calabi-Yau threefolds. I will also discuss some generalisation of the methods to hyper-surfaces in toric varieties.

For a compact quasi-smooth derived scheme M with (-1)-shifted cotangent bundle N, there are at least two ways to localise the virtual cycle of N to M via torus and cosection localisations, introduced by Jiang-Thomas. We produce two virtual cycles on the projective completion of N and one on the projectivisation of N; and show the ones on the completion push down to Jiang-Thomas cycles and the one on the projectivisation computes the difference. Using similar ideas we give an expression of the difference of the quintic and t-twisted quintic GW invariants of Guo-Janda-Ruan.

Bridgeland stability conditions are a generalization of the classical notion of slope stability for vector bundles on curves. However, it is in general a subtle problem to construct stability conditions on varieties in dimensions greater than 3. Building on work of Liu yielding stability conditions on products of curves, along with input from general theory on equivariant categories, we produce stability conditions on a certain family of strict Calabi-Yau varieties of arbitrary dimension by proving a case of the derived McKay correspondence. This is joint work with Alexander Perry.

Equivariant elliptic cohomology is an equivariant cohomology theory associated to an elliptic curve. After a review of its complexified incarnation, introduced by Grojnowski in 1994, we will give a new perspective on this invariant which involves derived mapping stacks. This naturally gives rise to an algebraic avatar of equivariant elliptic cohomology, that we call (equivariant) elliptic Hochschild homology. Based on joint work with Nicolò Sibilla.