Beijing Institute of Mathematical Sciences and Applications

Smooth projective Gm-varieties with isolated rational fixed points admit Tate Milnor-Witt motives. Over Euclidean fields, we give a splitting for- mula of such motives, which reduces the computation their Chow-Witt groups to that of their Chow groups and cohomologies of Witt sheaf. This a joint work with Jean Fasel.

It was conjectured by McKernan and Shokurov that given a Fano fibration from X to Z, the singularities on Z are bounded in terms of those on X. Recently this conjecture was proved by Birkar. In this talk, I will discuss explicit bounds of singularities on Z in this conjecture in relative dimension one and in the toric case.

In this talk, I will present our recent work concerning the explicit upper bound for anti-canonical volumes of e-lc Fano threefolds. More generally, We provide a reasonably small explicit upper bound with a sharp order, for the anti-canonical volumes of threefolds of e-lc Fano type (0＜e＜1/3). This is a joint work with Chen Jiang.

Let $X$ be a compact complex manifold in Fujiki's class C, i.e., admitting a big $(1,1)$-class $[\alpha]$. Consider $Aut(X)$ the group of biholomorphic automorphisms and $Aut_{[\alpha]}(X)$ the subgroup of automorphisms preserving the class $[\alpha]$ via pullback. We show that X admits an $Aut_{[\alpha]}(X)$-equivariant Kähler model. I will talk several applications. We show that $Aut_{[\alpha]}(X)$ is a Lie group with only finitely many components, which generalises an early result of Fujiki and Lieberman on the Kähler case. We also show that every torsion subgroup of $Aut(X)$ is almost abelian, and $Aut(X)$ is finite if it is a torsion group. This is based on a joint work with Sheng Meng.

“Quantum Lefschetz” is a pretentious name for understanding how moduli spaces -- and their virtual cycles and associated invariants -- change when we apply certain constraints. (The original application is to genus 0 curves in P^4 when we impose the constraint that they lie in the quintic 3-fold.) When it doesn’t work there are fixes (like the p-fields of Guffin-Sharpe-Witten/Chang-Li) for special cases associated with curve-counting. We will describe joint work with Richard Thomas developing a general theory.

We elaborate on construction of a derived Lagrangian intersection theory on moduli spaces of divisors on compact Calabi Yau threefolds. Our goal is to compute deformation invariants associated to a fixed linear system of divisors in CY3. We degenerate the CY3 into a normall crossing singular variety composed of Fano threefolds meeting along a K3. The deformation invariance arguments, together with derived Lagrangian intersection counts over the special fiber of the induced moduli space degeneration family, provides one with invariants of the generic CY fiber. This is report on several joint projects in progress with Ludmil Katzarkov, Tony Pantev, Vladimir Baranovsky and Maxim Kontsevich.

In this talk I will review few conjectures which aim to detect which linear differential equations come from Gauss-Manin connections, that is, they are satisfied by periods of families of algebraic varieties. This includes conjectures due to Katz-Grothendieck, André and Bombieri-Dwork. I will discuss another finer criterion to detect differential equations coming from families of hypergeometric Calabi-Yau varieties. Finally, I will explain a classification list in the case of Heun and Painlevé VI equations (joint works with S. Reiter).

After mentioning some backgrounds, I'll talk about a blowup formula for sheaf-theoretic virtual enumerative invariants on projective surfaces and its applications at the level of the generating series of those invariants. For instance, we obtain Goettsche-Kool's conjectural blowup formulae for the generating series of the virtual Euler characteristics and virtual $\chi_y$-genera of the moduli spaces, in which modular forms appear in the same way as in Vafa-Witten's original paper in '94. This determines some of universal functions in the generating series of Vafa-Witten invariants on a projective surface, which were conjectured also by Goettsche-Kool and Goettsche-Kool-Laarakker. These are based on joint work arXiv:2107.08155 with Nikolas Kuhn and arXiv:2205.12953 with Nikolas Kuhn and Oliver Leigh.

The Donovan-Wemyss conjecture from 2013 states that a compound Du Val singularity is determined up to isomorphism by the derived equivalence class of the "contraction algebra" associated with any crepant resolution. We will review the conjecture and its recent proof based on previous work by Wemyss, August, Hua-K and on the derived Auslander-Iyama correspondence, a deep result obtainedin August 2022 by Gustavo Jasso and Fernando Muro.

In this talk we will discuss recent results regarding singularities on fibrations, in particular, Fano and Calabi-Yau fibrations. We will give some of the main ideas of the proofs of these results.

I will report the establishment of the minimal model program for algebraically integrable foliations on klt varieties and it applications, such as the minimal model program for generalized pairs and the canonical bundle formula. If time permits, I will discuss some related open problems and their connections to moduli theory. This talk is partially based on a series of joint works of myself with Guodu Chen, Jingjun Han, Fanjun Meng, and Lingyao Xie.

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I will report on a joint work with Yuri Tschinkel (Simons Foundation) and Zhijia Zhang (Courant Institute) on linearizability of actions of finite groups on singular cubic threefolds.

In this talk, I will survey some recent progress about Kollar’s injectivity theorem on cohomologies. In the category of generalised pairs, we show that the injectivity theorem holds for projective generalised pairs with mild singularities when the nef part of the generalised pair is b-abundant. Moreover, by applying the methods from complex analytic geometry, we show that the injectivity theorem holds for generalised Kawamata log terminal generalised pairs on projective surfaces without any extra conditions on the nef parts of the generalised pairs.

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