2024年12月13-16日, 同济大学数学科学学院
胡飞（南京大学）
刘济豪（北京大学）
刘杰（中科院数学与系统科学研究院）
吕人杰（厦门大学）
欧文浩（中科院数学与系统科学研究院）
曲三太（中国科学技术大学）
孙浩（华南理工大学）
魏传豪（西湖大学）
12月13日
报到
12月14日
9:00-10:00 胡飞
合影
10:30-11:30 曲三太
午餐
14:00-15:00 魏传豪
15:20-16:20 孙浩
16:40-17:40 吕人杰
晚宴
12月15日
8:30-9:30 刘杰
9:45-10:45 欧文浩
11:00-12:00 刘济豪
午餐
12月16日
自由讨论
胡飞: An upper bound for polynomial volume growth of automorphisms of zero entropy
Let X be a smooth complex projective variety of dimension d and f an automorphism of X. Suppose that the pullback f^* of f on the real Néron–Severi space N^1(X)_R is unipotent and denote the index of the eigenvalue 1 by k+1. We prove an upper bound for the polynomial volume growth plov(f) of f, or equivalently, for the Gelfand–Kirillov dimension of the twisted homogeneous coordinate ring associated with (X, f), as follows: plov(f) \leq (k/2 + 1)d. Combining with the inequality k \leq 2(d-1) due to Dinh–Lin–Oguiso–Zhang, we obtain an optimal inequality that plov(f) \leq d^2, which affirmatively answers questions of Cantat–Paris-Romaskevich and Lin–Oguiso–Zhang. This is joint work with Chen Jiang.
曲三太: Irrationality of degenerations of Fano varieties
In this talk I will introduce a result about bounding degrees of irrationality of degenerations of klt Fano varieties of arbitrary dimensions. This proves the generically bounded case of a conjecture proposed by C. Birkar and K. Loginov for log Fano fibrations of dimensions greater than three. Our approach depends on a method from logarithmic geometry to modify the klt Fano fibration to a toroidal morphism of toroidal embeddings with bounded general fibres. This is a joint work with Prof. C. Birkar.
魏传豪: On the Hodge theory of Toroidal embeddings and corresponding Vanishings
Deligne's logarithmic comparison theorem and the degeneracy of the spectural sequence of logarithmic de Rham complex gives the mixed Hodge structure of a projective smooth variety with a normal crossing boundary divisor. In this talk, we will try to build a similar theory on toroidal embeddings. In particular, we will show the E_1-degeneracy of the spectral sequence of the logarithmic de Rham complex of any toroidal triple. This gives a geometric proof of a more general version of Danilov's conjecture.
孙浩: Parahoric Group Schemes on Higher dimensional Varieties
Parahoric group schemes G on curves are introduced by Bruhat and Tits. As an analogue of parabolic bundles, Balaji and Seshadri prove that there is a one-to-one correspondence between parahoric G-torsors on curves and equivariant principal bundles on appropriate covers. Recently, Balaji and Pandey introduced and studied parahoric group schemes on higher dimensional varieties. In this talk, I will discuss this result, and if time permits, I will say how it relates to nonabelian Hodge correspondence. This is joint work with Mao Sheng and Jianping Wang.
吕人杰: Generic Triviality of Automorphisms of Complete Intersections
The automorphism group is an important invariant for algebraic varieties. Our project studies the behaivor of automorphism groups for a family of smooth polarized varieties. With certain conditions on the moduli stack and the monodromy group associated to the family, we show that the general automorphism group is minimal. In particular, in most cases, a general smooth complete intersection has no non-trivial automorphisms. This is a joint work with Dingxin Zhang.
刘杰: Symplectic Singularities Arising from the Algebra of Symmetric Tensors
The symplectic singularities are introduced by Arnaud Beauville in 2000 as a local singular analogue of hyperkähler manifolds. There are several constructions for such kind of singularities in the literature, e.g. symplectic quotients, closures of nilpotent orbits and symplectic reductions etc. In this talk, I will present a new construction of symplectic singularities by using a kind of generalized Springer maps and the relation of this new construction with the previous ones will also be discussed. This is joint work with Baohua Fu.
欧文浩: Orbifold modification of complex analytic varieties
We prove that if $X$ is a compact complex analytic variety, which has quotient singularities in codimension 2, then there is a projective bimeromorphic morphism $f\colon Y\to X$, such that $Y$ has quotient singularities, and that the indeterminacy locus of $f^{-1}$ has codimension at least 3 in $X$. As an application, we deduce the Bogomolov-Gieseker inequality on orbifold Chern classes for stable reflexive coherent sheaves on compact Kaehler varieties which have quotient singularities in codimension 2.
刘济豪: Adjoint Foliated Structures on Surfaces
The concept of adjoint foliated structures is a recently introduced concept in the study of foliations, which characterizes both the structure of the foliation itself and the ambient variety. In this talk, I will discuss recent (and ongoing) work on adjoint foliated structures on surfaces, as well as related problems in explicit birational geometry of potential interest. This talk is based on joint works with Paolo Cascini, Jingjun Han, Fanjun Meng, Calum Spicer, Roberto Svaldi, and Lingyao Xie.
同济大学数学科学学院，衷和楼409报告厅
金方舟, 李灵光, 林胤榜, 张希平, 张子立, 朱子文
组织单位：同济大学数学科学学院
资助项目：科技部国家重点研发计划项目2021YFA1001400、国家自然科学基金青年项目12101455