Algebraic Geometry Seminar

Tautological relation on the moduli space of stable curves were studied by several method. I will reveiw the method used by Pandharipande-Pixton to prove the tautological relation on the moduli space of stable curvse using stable quotient. I will explain how to extend the result to get tautological relations on relative Picard stack over the moduli space of stable curves by constructing the stable quotient over Picard stack. After this I will explain how to get the result on the Picard stack which extends the original form of tautological relation given by Faber and Zagier. This talk is based on the joint work in progress with Younghan Bae.

To the singularities of a hypersurface one may associate increasingly sophisticated invariants: Milnor numbers, sheaves of vanishing cycles and categories of Matrix factorizations. It is an interesting problem to globalize these locally defined invariants. The Milnor numbers associated to the moduli spaces of suitable sheaves on a Calabi-Yau 3-fold leads to the highly influential Donaldson-Thomas invariants. Brav-Bussi-Dupont-Joyce-Szendroi have globalized the sheaf of vanishing cycles to construct a categorification of Donaldson-Thomas invariants (obstructed by some orientation data). I will take about globalizing higher categorical invariants. This depends on the study of (-1)-shifted symplectic structures on derived schemes, but I will not expect the audience to know what those words mean. This is work in progress, joint with B. Hennion and M. Robalo.

I will explain my latest joint work with Tsai, Vilonen and Xue (arXiv:2409.04030) about the nearby-cycle construction of cuspidal character sheaves on graded Lie algebras. These objects are closely related to finite-dimensional representations of double affine Hecke algebras and expected to play a role in the harmonic analysis on symmetric spaces and p-adic groups. I will start by explaining the more classical theory of sheaves on reductive Lie algebra and its Z-graded version. I will then highlight the new phenomena which occur when finite-order gradings on Lie algebras come into play.

Wall-crossing formulae for sheaf-counting invariants were recently reformulated by Joyce in terms of natural-looking vertex algebras living on homologies of moduli stacks. One of the downsides is the necesity to work with the full virtual fundamental classes instead of concrete invariants. To remedy this, I introduce additive formal families of vertex algebras that appear, for example, once one includes tautological insertions. In the second half of the talk, I will apply the general framework to conjectures about stable pair invariants while leaving a more detailed discussion for the third talk.

I will start by introducing some notation for Calabi-Yau dg-quivers. This language comes in handy once one starts talking about the obstruction theories of enhanced master spaces for sheaves. These can be constructed explicitly by geometric arguments, and are the key to proving wall-crossing formulae from the previous talk. Because the invariants counting semistable sheaves need to be defined even in the presence of strictly semistable, I will finish with the proof that this definition is independent of choices.

Any unfinished proofs from the second talk will be presented. I will then come back to the wall-crossing for pairs and discuss it in more detail. For example, I will explain why Joyce—Song wall-crossing holds, and how far one can get in proving Fano 3-fold DT/PT correspondences using formal families of vertex algebras.

I'll discuss a certain class of sheaves admitting an action of vector fields by Lie derivatives. These sheaves include D-modules, but there are a great many more natural examples, such as any sheaf of jets of sections of (p,q)-tensors. I'll then introduce a theorem, joint with Henrique Rocha, explaining how these sheaves are in a sense a bounded distance away from D-modules. I will formulate the theorem in a more geometric fashion, with a focus on the Atiyah algebra - the algebra of infinitesimal symmetries of a pair consisting of a space and a bundle on it, and actually prove a strengthened form of it in the complex analytic setting.

This is the first of a series of two talks. In this talk I intend to provide a reasonably in depth introduction to vertex algebra theory, assuming minimal background. Having set up the necessary prerequisites, I will in particular go over the construction the chiral de Rham complex, a sheaf of vertex algebras on any smooth variety and explain the supersymmetry such a sheaf inherits when the variety is Calabi-Yau. Relations to the two variable elliptic genus will also be discussed. The second talk will deal with various non-commutative generalisations of the above, arising in some of my work.

After reviewing the theory of local systems on homotopy types and describing the similarities and differences with quasi-coherent sheaves on schemes, we study the case of local systems on the classifying space BG of a compact Lie group G of dimension d and give conditions for the existence of a proper Calabi-Yau structure of dimension -d. By the solution of the cyclic Deligne conjecture or the 2 dimensional cobordism hypothesis, we then obtain string topology operations on the cochains of the free loop space LBG. This is joint work with N. Rozenblyum.

Symplectic cohomology is a powerful invariant associated to open symplectic manifolds. It is essential in modern symplectic dynamics, it is closely related to quantum cohomology, and it plays an important role in mirror symmetry. However, it is not very sensitive to the homotopy type of the underlying manifold: it can even vanish for manifolds with arbitrarily complicated topology. When some extra structures on it are remembered, the rational homology can be recovered as a variant of Tate cohomology, but the torsion information is completely lost. In this talk, I will explain how to recover further information, including torsion part of the homology, complex K-theory and more from an enhanced version of symplectic cohomology. This is joint work with Laurent Cote.

In joint work with Mark Gross, we show that punctured log Gromov--Witten invariants of Abramovich--Chen--Gross--Siebert, in the case of log Calabi--Yau varieties obtained by blows ups of toric varieties along hypersurfaces on the toric boundary, can be captured from a combinatorial algorithm encoded in a scattering diagram. This is an extension of the previous work of Gross--Pandharipande--Siebert to higher dimensions.

Quiver Donaldson-Thomas invariants are integers determined by the geometry of moduli spaces of quiver representations. They play an important role in the description of BPS states of supersymmetric quantum field theories. I will describe a correspondence between quiver Donaldson-Thomas invariants and Gromov-Witten counts of rational curves in toric and cluster varieties. This is joint work with Hülya Argüz (arXiv:2302.02068 and arXiv:2308.07270).

After a very brief vertex algebra recap I'll introduce the chiral de Rham complex and go over its basic properties, such as modularity of characters in the Calabi Yau case. I'll then sketch the construction of some noncommutative avatars of the above, arising in some of my work. I'll pay particular attention to the case of vanishing cycles type constructions.

We discuss some results on genus 1 Gromov-Witten invariants of Hilbert scheme of points on the affine plane, including a determination of multi-point series in terms of one-point series and a close formula for an one-point series.

In a 2018 Physics paper, Jockers and Mayr argued that a so-called “3-d, N=2 abelian Gauged Linear Sigma Model” leads, via the renormalization flow, to Givental’s permutation-equivariant quantum K-theory of a complete intersection in a toric variety. Among the consequences they explored was a new derivation of the integrality of the Taylor coefficients of the mirror map (for Calabi-Yau complete intersections) which leveraged the intrinsic integrality of the 3-d GLSM partition function. In this talk I will explain how the theory of quasimaps to GIT quotients can be used to justify rigorously their argument (and extend its applicability to the non-abelian case as well). This is joint work with Jorin Schug.

In the series of two talks, I will review the notation of stable quasimaps and their wall-crossing, introduce the master space technique and prove the wall-crossing formula, discuss some applications and variants of quasimap wall-crossing, discuss Mixed-Spin-P fields and mathematical Gauged Linear Sigma Models, which can also be viewed as generalizations of quasimaps.

In this joint work with Alina Marian, we study the geometric representation theory of Quot schemes of 0-dimensional coherent sheaves on curves: the overall structure we obtain is a categorification of quantum loop sl_2 acting on the derived categories of Quot schemes. We have two uses for this machinery: we produce an explicit semiorthogonal decomposition of the derived categories in question (which we expect match Toda's wall-crossing construction), and we prove conjectures of Krug and Oprea-Sinha on the cohomology of tautological bundles on Quot schemes.

I'll talk about joint work with Adeel Khan, where we define an elliptic version of algebro-geometric loop spaces. I'll recall the rational and trigonometric stories in some detail before explaining our construction in the elliptic case.

Gromov-Witten invariants of a projective variety X come from enumerating curves in X. A famous example for X is the quintic hypersurface in P^4, studied by string theorists regarding mirror symmetry. Invariants from genus zero curves into X have been computed out of invariants of P^4, following the so-called quantum Lefschetz principle. However, invariants from higher genus curves are much harder to compute. In particular, based on a specific example, one may believe that they cannot be deduced from invariants of P^4. In this talk, I will explain my ongoing work in which I am looking for this relation, presenting a conjectural formula, where it comes from and a possible road to proving it.

I will report on joint work with Yukinobu Toda (partially in progress) about the derived category of coherent sheaves of semistable Higgs bundles on a curve.

These categories have semiorthogonal decompositions in certain categories analogous to the "window categories" of Halpern-Leistner, Ballard-Favero-Katzarkov, Špenko-Van den Bergh. In the first half of the talk, I will discuss the general theory of "window categories".

Next, I will focus on two conjectural dualities. The first is between semistable Higgs bundles of degree zero and a "limit" category. This equivalence aims to make precise the proposal of Donagi-Pantev of considering the classical limit of the de Rham Langlands equivalence. The second is a primitive version of the first, and it relates categories of sheaves on moduli of semistable Higgs bundles (for various degrees). This equivalence may be regarded as a version of the D-equivalence conjecture / SYZ mirror symmetry. We can prove (partial) versions of these conjectures for topological K-theory of these categories.

Motivated by strange duality, Johnson predicted a correspondence between Segre and Verlinde invariants defined for Hilbert schemes of points on surfaces. In my previous work, I studied this correspondence for Quot schemes parametrizing zero-dimensional quotients of vector bundles on curves, surfaces and fourfolds. All of these results have been formulated for compact geometries only. Motivated by this, we have looked at the fully equivariant case with Huang Jiahui. In the talk, I will discuss the issues posed by non-zero degree equivariant contributions and how the formulation generalizes to this setting.

Let G be an algebraic group acting on a scheme X. Then its equivariant algebraic K-theory maps into global invariant functions on a certain associated fixed-point scheme, which is of independent interest in geometric representation theory. After recalling this comparison and its relationship to equivariant Hochschild homology, I will explain what simplifies in characteristic p after perfection. The resulting map is an equivalence in interesting cases, such as torus-equivariant algebraic K-theory of split toric varieties or affine Schubert varieties in the perfect GL_n-affine Grassmannian.