Titles and Abstracts

Title: The moduli space of abelian varieties and its tautological ring

Abstract: We survey recent results on algebraic cycles on the moduli space of principally polarized abelian varieties . In particular, we discuss the tautological ring, the decomposition of cycle classes into tautological and non-tautological components via the tautological projection, and explicit examples of non-tautological classes. A parallel theory can be developed for the universal abelian variety. This work is based on joint projects with Samir Canning, Lycka Drakengren, Jeremy Feusi, Daniel Holmes, Aitor Iribar Lopez, Denis Nesterov, Rahul Pandharipande, Johannes Schmitt, and Zheming Sun.

Title: On the P=C conjecture

Abstract: The singular cohomology of the moduli space of one-dimensional sheaves on a del Pezzo surface admits two filtrations: the perverse filtration, which records the topology of the Hitchin-type fibration, and the Chern filtration, which reflects the shape of tautological relations. The P=C conjecture asserts that the two filtrations, despite their very different nature, coincide. I will discuss a result on the P=C conjecture for the top cohomological degree, and a certain chi-independence phenomenon. This is joint work in progress with M. Moreira, W. Pi.

Title: Mixed-Spin-P fields and GLSM

Abstract: The theory of Mixed-Spin-P (MSP) fields was introduced by Chang-Li-Li-Liu for the quintic threefold, aiming at studying its higher-genus Gromov-Witten invariants. Chang-Guo-Li has successfully applied it to prove conjectures including the BCOV Feynman rule, Yamaguchi-Yau's polynomiality conjecture and the Holomorphic Anomaly Equation. In this talk, we generalize the construction of MSP to a more general setting, including complete intersections in toric DM stacks. The key is to introduce a new stability condition for the moduli of "fields". This also generalizes the construction of mathematical Gauged Linear Sigma models by Fan-Jarvis-Ruan. The talk is based on a joint work with Huai-Liang Chang, Shuai Guo, Jun Li and Wei-Ping Li.

Title: Moduli Spaces of Quasisections and Vafa-Witten Invariants on Elliptic Surfaces

Abstract: For a threefold X = C x S, D. Nesterov developed a theory of quasimaps from the curve C to the moduli space of sheaves on the surface S. He compared the moduli space of quasimaps with the moduli space of sheaves on the threefold X, as well as their obstruction theories. As a special case, when C is an elliptic curve and S is the total space of a smooth projective curve C', he computed the Vafa-Witten invariants of the surface C \times C' using the theory of quasimaps.

When X is a smooth fibration over C, I will explain how to generalize Nesterov's theory using the notion of quasisections—which are modifications of quasimaps—and establish the correspondence between the moduli space of sheaves on X and the moduli space of quasisections into the relative moduli space of sheaves. As an application, I will explain how to compute the Vafa-Witten invariants of an elliptic surface that is a smooth fibration over C. For example, we can consider the case where S = [C x E / G], with a finite group G acting freely on a curve C and an elliptic curve E.

Moreover, for the case where the elliptic surface has nodal fibers, we will discuss how to generalize the theory of quasisections. This is based on joint work in progress with Yaoxiong Wen.

Title: Singularities and syzygies of secant varieties of smooth projective curves

Abstract: We’ll report on joint work with Wenbo Niu and Jinhyung Park. Let C be a smooth projective curve of genus g and L be a line bundle of degree d line bundle on C, where d >= 2g+2k+p+1.Let Sigma_k(X, L) be the k th secant variety of C in P^r = P(H^0(L)). We show that we can control first p syzygies of the coordinate ring of Sigma_k(X, L). Furthermore the k th secant variety is projectively Cohen Macaulay with Bois singularities.

Title: Derived Isogeny between Ricci-flat Varieties

Abstract: Derived isogeny on algebraic varieties is a natural generalization of derived equivalence and captures new invariants. This talk aims to elucidate the concrete geometric implications of this new equivalence relation on Ricci-flat varieties—particularly on hyper-Kähler varieties and abelian varieties. We will systematically explore the resulting classification problems and related properties. This is based on joint works with Z. Lu, Z. Tang and R. Zhang.

Title: Ulrich bundles on the intersection of two quadrics

Abstract: Ulrich bundles are special vector bundles on algebraic varieties drawing lots of attention these days. In the first part of this talk, I will briefly review some basic definitions and results about Ulrich bundles. Then I will discuss Ulrich bundles on the intersection of two quadrics. The last part of this talk is based on a joint work in progress with Jiwan Jung and Han-Bom Moon.

Title: Curves on K3 Surfaces

Abstract: A folklore conjecture in algebraic geometry, sometimes attributed to Bogomolov, states that every complex algebraic K3 surface contains infinitely many rational curves. For generic K3 surfaces, this was shown by Xi Chen in 1998. Important special cases were obtained in work of Bogomolov and Tschinkel, who established it, among other cases, for K3 surfaces of Picard rank greater or equal to 5. Bogomolov, Hassett, and Tschinkel proposed a strategy to deal with the remaining cases in 2011 and in joint work with Jun Li in 2012, we used this strategy to establish the conjecture for K3 surfaces of odd Picard rank, which includes the rank 1 case. The remaining cases were finally dealt with in 2022 in joint work with Xi Chen and Frank Gounelas, where we also generalise and establish this conjecture in higher genus. In my talk, I will give an overview of these results and their proofs.

Title: Instanton moduli spaces on the blowup surfaces and affine vertex algebras.

Abstract: Vafa–Witten observed that Yoshioka’s blow-up formula for the Euler characteristics of rank 2 instantons on an algebraic surface coincides with the character of the Wess–Zumino–Witten model for SU(2) at level 1, and raised the question of finding a rational conformal field theory explanation for this striking coincidence. In the joint work with ZHAO Yu and JIANG Qingyuan, we provide an answer to this question by constructing and analyzing the affine glr action on various cohomology theories, including the Grothendieck group of coherent sheaves, Hochschild homology, Chow groups, and Hodge cohomology, of the moduli space of stable sheaves on a blown-up surface.

Title: Lagrangian classes

Abstract: In this talk, I will explain the construction of Lagrangian classes for perverse sheaves in cohomological Donaldson-Thomas theory, whose existence was conjectured by Joyce. The two key ingredients are a relative version of the DT perverse sheaves and a hyperbolic version of the dimensional reduction theorem. As a special case, we recover Borisov-Joyce/Oh-Thomas virtual classes in DT4 theory.

As applications, I will explain how to construct the following structures from the Lagrangian classes: (1) cohomological field theories for gauged linear sigma models, (2) cohomological Hall algebras for 3-Calabi-Yau categories, (3) relative Donaldson-Thomas invariants for Fano 4-folds with anti-canonical divisors, (4) refined surface counting invariants for Calabi-Yau 4-folds.

This is joint work in progress with Adeel Khan, Tasuki Kinjo, and Pavel Safronov.

Title: Properness of K-moduli

Abstract: (Joint with Harold Blum, Yuchen Liu, and Ziquan Zhuang) We present a new proof of the properness of K-moduli spaces. While our approach still depends on the higher-rank finite generation theorem, it avoids the use of Halpern-Leistner’s Θ-stratification theory. Instead, we develop a purely birational method, rooted in a relative framework for K-stability, which provides a more direct geometric proof of properness.

Title: Boundedness for K-trivial varieties with fibrations

Abstract: According to the Beauville-Bogomolov decomposition theorem, any smooth K-trivial variety admits a finite cover by a product of (1) abelian varieties, (2) strict Calabi-Yau varieties, and (3) irreducible holomorphic symplectic varieties (IHSV). In a fixed dimension, all abelian varieties are diffeomorphic, and indeed deformation equivalent through non-algebraic complex tori. The corresponding question remains largely open for cases (2) and (3). If we assume that a variety of class (2) or (3) admits a non-trivial fibration structure, then much more can be shown. In particular, fibered Calabi-Yau threefolds have bounded moduli problem, and IHSV of a fixed dimension with a Lagrangian fibration have bounded moduli. This is based on joint work with Engel, Filipazzi, Mauri, and Svaldi.

Title: Categorical Donaldson-Thomas theory and Dolbeault Langlands duality

Abstract: I will review Donaldson–Thomas invariants of Calabi–Yau threefolds, their cohomological refinements, and their generalizations to BPS invariants. I will then explain motivations for categorifying these invariants, in particular from the viewpoint of categorical wall-crossing formulas and the construction of (twisted) categorical crepant resolutions of singularities. Finally, I will discuss how categorical DT theory leads to a precise formulation of the Dolbeault geometric Langlands conjecture of Donagi–Pantev, an equivalence of categories for moduli stacks of Higgs bundles on curves (joint work with Tudor Padurariu). I will prove this conjecture for GL_2 over the locus in the Hitchin base corresponding to reduced spectral curves.

Title: Virtual cycles via Fulton classes

Abstract: The virtual cycle of a quasi-smooth scheme coincides with that of its (-2)-shifted cotangent bundle. For a derived scheme whose tangent complex has three terms -- which may be regarded as the mildest non-quasi-smooth case -- we define the equivariant virtual cycle using the Fulton class, then prove the analogous result in the equivariant setting.

A non-equivariant cycle can also be defined, but it need not coincide with that of the (-2)-shifted cotangent bundle. We explain how the two differ.

Title: On non-abelian Fourier tranforms for compact hyper-Kähler varieties.

Abstract: Compact hyper-Kähler varieties are higher dimensional analogs of K3 surfaces. It has been expected that, from many perspectives, the geometry of compact hyper-Kähler varieties "behaves like" that of abelian varieties (e.g. the Beavuille-Voisin conjectures predict that this is the case for motives and algebraic cycles.) In this talk, I will discuss this phenomenon from the perspective of Fourier transforms. I will explain that for hyper-Kähler varieties of K3[n]-type, the analogues of Makai's Poincaré line bundles for abelian varieties should be given by certain hyperholomorphic bundles constructed by Markman recently. These ideas lead to a proof of the Bondal-Orlov D-equivalence conjeture for K3[n]-type varieties (joint with Davesh Maulik, Qizheng Yin, and Ruxuan Zhang), and a proof of Fu-Vial's multiplicative version of the Orlov conjecture for K3[n]-type varieties for homological motives (joint work with Davesh Maulik and Qizheng Yin). Conjectures regarding this Fourier transform package will be discussed.

Title: Naive homotopical techniques in algebraic geometry

Abstract: I’ll present a framework for making simple but effective homotopical arguments in algebraic geometry. These methods apply only in certain fortunate situations, but when they do, they yield surprising (to me) facts about basic geometric objects. My goal is to share clear, concrete examples—proofs you can follow and take home. This is based on work with Hannah Larson, Jim Bryan, and Ben Church, and connects to ideas developed by many others.