Algebraic Geometry Seminar
Title: Hodge ideals and roots of the Bernstein-Sato polynomial
Speaker: Bingyi Chen (Tsinghua University)
Abstract: Hodge ideals, which are developed by Mustata and Popa, are important invariants of hypersurface singularities that arise naturally from Saito's theory of mixed Hodge modules. There are two diffierent approaches to study Hodge ideals: birational geomerty method by means of log resolution and D-module method by means of V-filtration. The theory of Hodge ideals connects these two diffierent fields and leads to a number of striking applications. In this talk, I will introduce the theory of Hodge ideals and talk about its application on the bound of roots of reduced Bernstein-Sato polynomial.
Title: Kohn-Rossi cohomology, complex Plateau problem and Rigidity of CR morphisms
Speaker: Stephen Yau (Tsinghua University)
Date: 2021-9-23Abstract: In this talk, we investigate the relationship between compact strongly pseudoconvex CR manifolds and the singularities of their Stein fillings. We first compute the dimensions of Kohn-Rossi cohomology groups with values in holomorphic vector bundles in terms of local cohomology groups. As an application, we solve the classical complex Plateau problem for compact strongly pseudoconvex CR manifolds when its Stein fillings has only isolated complete intersection singularities. Finally, using Kohn-Rossi cohomology, we obtain some sufficient conditions for the non-existence of CR morphisms between the links of isolated complete intersection singularities with different embedding codimensions. This is a joint work with Xiankui Meng.
Title: Measure the positivity of tangent bundles of Fano varieties
Speaker: Jie Liu (Chinese Academy of Sciences)
Abstract: While the properties of the anticanonical divisor $-K_X$ of a Fano variety $X$ and its multiples have been studied by many authors, the positivity of the tangent bundle $T_X$ is much more elusive. In this talk, I will introduce some invariants to measure the positivity of the tangent bundles of Fano varieties via their anti-canonical bundles and then I will discuss various interesting examples to illustrate the general situations. This is based on joint works with Baohua Fu, Andreas Höring and Feng Shao.
Title: Explicit boundedness of canonical Fano 3-folds: known results and open problems
Speaker: Chen Jiang (Fudan University)
Abstract: Motivated by the classification of canonical Fano 3-folds, we are interested in boundedness results on diffrent kinds of canonical Fano 3-folds, such as anticanonical systems, indices, degrees, and so on. I will summarize known results with some progress (based on joint works with Meng Chen and Yu Zou) and open problems in this area.
Title: Recent progress towards Shokurov’s ACC conjecture for mld
Speaker: Dr. Jihao Liu, Northwest Univ, US
Abstract: Shokurov's ascending chain condition (ACC) conjecture for minimal log discrepancies (mlds) is a core conjecture in birational geometry and has a deep relationship with the minimal model program. In particular, in 2004, Shokurov shows that the ACC conjecture and the lower-semicontinuity conjecture for mlds will imply the termination of flips and will thus complete the minimal model program. In this talk, I will discuss some recent progress towards Shokurov's ACC conjecture and some related questions. Some parts of this talk are joint works with Jingjun Han, V.V. Shokurov, Liudan Xiao, and Lingyao Xie.
Title: Sarkisov program and automorphism of affine spaces
Speaker: Yifei Chen (Chinese Academy of Sciences)