Beijing Institute of Mathematical Sciences and Applications

Kato homology is the homology of a special type of Gersten complex of (co)homology theories, first studied by Bloch-Ogus and Kato. Motivated by conjectures like geometric Manin conjecture, Cohen-Jones-Segal conjecture, and some previous work of myself and others on some arithmetic questions of rationally connected varieties defined over local and global fields, I will propose a conjecture about the Kato homology of rationally connected fibrations. One interesting feature is that the theory fits very well with the general framework of the minimal model program.

I will give an overview of an ongoing project with Vadim Vologodsky. It describes a “hidden” homogeneous structure of divided power envelopes in characteristic p. Applications include a geometric interpretation of p-curvature, a refinement of Mazur’s fundamental theorem relating the action of Frobenius to the Hodge and conjugate filtrations, and an explicit and purely crystalline construction of the Sen operator on mod p de Rham cohomology. As a consequence, we exhibit a close relationship between the Sen operator and the failure of “strong divisibility”.

Inspired by the Bombieri-Lang conjecture, we propose a program studying the loci of non-rigid maps into moduli spaces of varieties. We conjecture that if a "general'' moduli space is not birational to any Shimura variety of rank >1, then the loci of non-constant and non-rigid maps is contained in a proper subvariety of the moduli space. Under the assumption of a locally injective Torelli map, we find some evidence of this conjecture, which are consequences of the recent work by Baldi-Klingler-Ullmo on the distribution of Hodge loci. We are looking for a type of Ax-Schanuel statement for structurally atypical intersections. It is expected to play an important role in the program. This is a joint project with Ke Chen, Tianzhi Hu and Ruiran Sun.

In a series of papers, Lusztig developed a theory of characters of a reductive algebraic group G by using perverse sheaves on G. To get appropriate perverse sheaves on G (called character sheaves), he considered a family of subvarieties of the flag variety G/B parameterized by elements in G; now, we call Lusztig varieties. In this talk, we will explain how they are related to two interesting families of subvarieties of the flag variety, Schubert varieties and Hessenberg varieties. Regular semisimple Lusztig varieties share many nice properties with Schubert varieties. They are normal Cohen-Macaulay, have rational singularities, and are of Fano type. We construct a flat degeneration of regular semisimple Lusztig varieties to regular semisimple Hessenberg varieties and compare their cohomology spaces. This is joint work with P. Brosnan and D. Lee.

The relative Jacobian of a family of curves depends on the degree. When singular curves appear, the relative compactified Jacobian further depends on the choice of a stability condition. We discuss in this talk several dependence and independence results concerning the cohomology rings of universal (compactified) Jacobians. Notably, we construct a common degeneration of all cohomology rings of universal fine compactified Jacobians over the moduli of stable pointed curves, whose ring structure is independent of the degree or the stability condition. Joint work with Younghan Bae, Davesh Maulik, and Junliang Shen.

The local volume of a Kawamata log terminal (klt) singularity is an invariant that plays a central role in the local theory of K-stability. By the stable degeneration theorem, every klt singularity has a volume preserving degeneration to a K-semistable Fano cone singularity. I will talk about a joint work with Chenyang Xu on the boundedness of Fano cone singularities when the volume is bounded away from zero. This implies that local volumes only accumulate around zero in any given dimension.

A classical result of Newstead shows that the moduli space of rank-two stable vector bundles with fixed determinant of odd degree over a genus-two curve is isomorphic to a smooth three-dimensional complete intersection X of two quadrics. In this setting, the Hitchin morphism induces a Lagrangian fibration over the cotangent bundle of X. In this talk, I will present a generalization of this phenomenon to higher dimensions. The results are based on joint works with Arnaud Beauville, Antoine Etesse, Andreas Höring, Claire Voisin, and Vladimiro Benedetti.

One of the main open problems in the Minimal Model Program (MMP) is the termination. Motivated by local volumes introduced by Chi Li, we introduce log canonical volume which is non-decreasing in any sequence of MMP for general type varieties. As a result, in such kind of MMP, we show that (1) the Cartier index of any Weil Q-Cartier is uniformly bounded; (2) every fiber of the extremal contractions or the flips is bounded (3) the set of minimal log discrepancies belongs to a finite set. This is a joint work with Lu Qi, and Ziquan Zhuang.

Stein degree measures the number of connected components of general fibers of a projective morphism. A conjecture due to Caucher Birkar asserts that Stein degree of boundary divisors on log Calabi–Yau fibrations is bounded from above. In this talk, I will present our recent progress that establishes the boundedness of Stein degree, thereby confirming Birkar’s conjecture in the general framework of generalised pairs. This talk is based on joint work with Caucher Birkar.

Hyperbolicity is a key global property of moduli spaces of various algebraic varieties with non-negative Kodaira dimension. In this talk, I will introduce a natural stratification of the moduli stack of stable minimal models—introduced by Professor Birkar—including the moduli stack of KSBA pairs—such that the universal family over each stratum is equisingular in the sense of birational geometry. I then investigate the hyperbolicity properties of these strata. In particular, I will show various forms of hyperbolicity for the strata of the moduli stack $\overline{M}_{g,n}$ of stable curves with marked points, including both the open locus $M_{g,n}$ and the boundary strata associated with the boundary divisor $\partial \overline{M}_{g,n}$.

I will survey a recent work with J. Cao, namely arXiv:2502.02183. The main results concern properties of foliations with positive minimal slope. As application, given a compact Kähler manifold with pseudo effective canonical bundle we show that the determinant of any quotient of an arbitrary tensor power of the cotangent bundle is equally pseudo-effective.