Organizer：Caucher Birkar, Mao Sheng
Date: Oct.16-Oct18
Venue: Bimsa A6101
Zoom ID: 262 865 5007
Password：YMSC
Speakers
1. Jingjun Han (Fudan University)
2. Santai Qu（University of Science and Technology of China）
3. Baohua Fu（Chinese Academy of Science）
4. Mihai Paun (Universität Bayreuth)
5. Zuo Kang（Wu Han University）
6. Arthur Ogus（University of California, Berkeley）
7. Qizheng Yin（Peking University）
8. Zhiyu Tian（Peking University）
9. Jie liu（Chinese Academy of Science）
10. Zhuang Ziquan (John Hopkins)
Schedule
Oct.16th |
9:00-10:00 |
Zhiyu Tian |
Kato homology of rationally connected fibrations |
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. |
10:00-10:30 |
Tea Break |
10:30-11:30 |
Arthur Ogus |
Divided powers, p-curvature, and diffraction |
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.” |
11:30-13:30 |
Lunch |
13:30-2:30 |
Zuo Kang |
Loci of non-rigid families of varieties in the corresponding moduli space. |
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. |
14:30-15:00 |
Tea Break |
15:00-16:00 |
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16:00-16:30 |
Tea Break |
16:30-17:30 |
Qizheng Yin |
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17:30-20:00 |
Dinner |
Oct.17th |
9:00-10:00 |
ZiQuan Zhuang |
Boundedness of singularities and discreteness of local volumes |
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. |
10:00-10:30 |
Tea Break |
10:30-11:30 |
Jie Liu |
Intersection of two quadrics and the Hitchin morphism |
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. |
11:30-13:30 |
Lunch |
13:30-14:30 |
JingJun Han |
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14:30-15:00 |
Tea Break |
15:00-16:00 |
Santai Qu |
Stein degree on log Calabi-Yau fibrations |
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. |
16:00-16:30 |
Tea Break |
16:30-17:30 |
Junchao Shentu |
Stratified Hyperbolicity of the Moduli Stack of Stable Minimal Models |
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 $\partial \overline{M}_{g,n}$. |
17:30-20:00 |
Dinner |
Oct.18th |
9:00-10:00 |
Mihai Paun |
Positivity of holomorphic tensors on compact Kähler manifolds |
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. |
10:00-11:30 |
Tea Break |
10:30-11:30 |
TBD |
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