Peking University Algebraic Geometry Seminar

We present an adjunction formula for foliations on varieties and we consider applications of the adjunction formula to the cone theorem for rank-one foliations and the study of foliation singularities. Joint work with C. Spicer.

对于一个光滑的代数簇 X，它的有界导出范畴上所有 Bridgeland 稳定性条件组成的集合上有一个自然的拓扑。这使得所有的稳定性条件构成了一个复流形 Stab(X)。取定一个 character v，人们可以通过研究刻画模空间 M_σ(v) 随稳定性条件 σ 在 Stab(X) 上的变化来获得很多有意义的信息和结果。

当 X 的维数为 3 时，对模空间 M_σ(v) 的 chamber 结构的刻画变得非常困难。一个重要的原因是 Stab(X) 的维数过大，即使是商掉一些无效维数后，也还余下复二维，并且 chamber 的边界也是由二次方程给定的。在本报告中，我会介绍一种将 Stab(X) 实约化的思路。通过这个实约化，我们将得到一个新的实流形 StaR(X)。M_σ(v) 的全部 chamber 结构可以在实二维空间 StaR_v(X) 上刻画，并且边界也全部是线性空间。如果时间允许，我将介绍关于 StaR(X) 结构的一些猜想。

The abundance conjecture is one of the most important conjectures in algebraic geometry, particularly in birational geometry. In characteristic 0, it was proven in the 1990s that the conjecture holds for log canonical pairs of dimension 3. In this talk, we will discuss recent progress toward proving the abundance conjecture for threefolds in positive characteristic.

Deuring gave a now-classical formula for the number of supersingular elliptic curves in characteristic p. We generalize this to a formula for the cycle class of the supersingular locus in the moduli space of principally polarized abelian varieties of given dimension g in characteristic p. The formula determines the class up to a multiple and shows that it lies in the tautological ring. We also give the multiple for g up to 4. Joint work with S. Harashita.

The minimal log discrepancy is an invariant of singularities related to the termination of flips. The ACC for minimal log discrepancies is still unknown in dimension three, and it is one of the most important remaining problems in the minimal model theory of threefolds. In the talk, I will explain a proof of the ACC for minimal log discrepancies on smooth threefolds. More introductory exposition will be given in my seminar talk at the Academy of Mathematics and Systems Science on the preceding day, October 15.

We discuss a conjecture on rank-2 vector bundles on curves that Kazhdan made in his plenary talk at ICM 2022.

Irreducible symplectic varieties are one of the three building blocks of varieties with Kodaira dimension zero, which are higher-dimensional analogs of K3 surfaces. Despite their rich geometry, there have been only a limited number of approaches to construct irreducible symplectic varieties. In this talk, I will introduce a general criterion for the existence of irreducible symplectic compactifications of non-compact Lagrangian fibrations, based on the minimal model program and the geometry of general fibers. As an application, I will explain how to obtain a 42-dimensional irreducible symplectic variety with second Betti number at least 24, which belongs to a new deformation type. Joint work with Yuchen Liu and Chenyang Xu.

Remarkable progress has been made in recent years in the minimal model theory for complex algebraic varieties. The first breakthrough was brought by Birkar, Cascini, Hacon and McKernan. In 2022, Fujino generalized their results to projective morphisms between complex analytic spaces. This is the first step of the minimal model theory in the complex analytic setting. In this talk, I will introduce recent progress of the minimal model theory for log canonical pairs in the complex analytic setting. The talk includes joint work with Makoto Enokizono.

A compact complex manifold is prime Fano if c₁(X) is positive and generates the second integral cohomology group. In dimension 3, there are 10 deformation types, with parameter g varying from 2 to 10 and 12, by Fano–Iskovskih. The first half are easy to describe but seem "deeply" irrational. The latter half are linear sections of various Grassmann varieties associated to Dynkin diagrams D₅, A₅, C₃, G₂ for g < 12, and deformations of the Umemura compactification of SL(2)/Icosa for g = 12. After a brief survey, I will explain a recent discovery on their relation with supersingular K3 surfaces, which are mostly obtained from the (Bring–)Reid sextic surface modulo 2, 3 and 5.

One of the main problems of birational geometry is the classification of algebraic varieties up to birational equivalence. Refining this problem, one can classify algebraic varieties with additional structure, for example varieties with a fixed (meromorphic) volume form. In this case, it is natural to consider volume forms with poles of at most first order. The group of equivalence classes of varieties with such a form is called the Burnside group. This group is good because some natural invariants of birational maps preserving the volume form on a given variety take values in it. We will define and study these invariants (sometimes called "motivic invariants") for groups of birational automorphisms of a projective space with a "standard" toric-invariant form. We will show that such groups are not simple in any dimension starting from four, and also that they cannot be generated by pseudo-regularizable elements. This result can be seen as a generalization of a similar theorem for the classical Cremona group, the group of birational automorphisms of the projective space.

It is anticipated that the Borisov–Alexeev–Borisov conjecture also holds in the context of foliations. To deepen our understanding, we examine Fano foliations within the toric category. In this talk, I will discuss the boundedness of the toric Fano adjoint foliated structure with mild singularities. Joint work with Chih-Wei Chang.

The abundance conjecture predicts that, on a minimal projective klt pair (X, D), the adjoint divisor K_X + D is semiample. In this talk, I will explain how currents, and in particular currents with minimal singularities, play a role in recent progress towards a proof of the abundance conjecture for minimal klt pairs with non-vanishing Euler–Poincaré characteristic.

A Calabi–Yau fibration is a fibration of projective varieties X → Z such that the canonical bundle K_X is numerically trivial over Z. This class of varieties plays a significant role in algebraic geometry, appearing naturally in contexts such as good minimal models and elliptic Calabi–Yau varieties. In this talk, I will present some results on the boundedness of Calabi–Yau fibrations under certain natural conditions, based on joint work with Minzhe Zhu and Xiaowei Jiang.