Algebraic Geometry Seminar

Abstract: Given a big line bundle L on a projective manifold, Lazarsfeld–Mustată and Kaveh–Khovanskii introduced method of constructing convex bodies associated with L. These convex bodies are known as Okounkov bodies. When L is endowed with a singular positive Hermitian metric h, I will explain how to construct smaller convex bodies from the data (L,h). These convex bodies play important roles in the study of the singularities of h.

Abstract: Foliation theory has been a crucial role in birational geometry over the past few decades, such as in Miyaoka's resolution of the abundance conjecture for complex projective varieties of dimension three. This talk aims to provide an overview of the theory of birational geometry of foliations and highlight some of its applications.
 

Abstract: The purpose of this talk is to provide an overview of the recent developments on the Minimal Model Program for algebraically integrable foliations over a complex projective variety. 
 

Abstract: Minimal model program for foliations is an analogue of the classical minimal model program. Foliated dlt foliations play the same role as dlt pairs in the classical minimal model program, making it a natural class of singularities to study in the theory of foliations. In this talk, we show that we can run an MMP on a Q-factorial foliated dlt algebraically integrable foliation. Numerous applications also will be presented. Joint work with Jingjun Han, Jihao Liu, and Lingyao Xie.

Abstract: 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.

Abstract: The concept of generalized pairs, introduced by Birkar and Zhang, plays an important role in modern birational geometry. This concept has been particularly crucial in proving the Borisov-Alexeev-Borisov conjecture and the McKernan-Shokurov conjecture. In this talk, I will explain how the theory of generalized pairs is applied to foliation theory from various perspectives, including but not limited to: Bertini-type theorems, the canonical bundle formula, the minimal model program (MMP) with scaling, the base-point-freeness theorem, the existence of flips, flops between minimal models, and the explicit adjunction formula. The talk is based on several joint works and ongoing collaborations with one or more of the following scholars: Paolo Cascini, Guodu Chen, Yifei Chen, Jingjun Han, Yujie Luo, Fanjun Meng, Calum Spicer, Roberto Svaldi, Yanze Wang, and Lingyao Xie.

Abstract: I will show the singularities of the secant variety to a smooth projective subvariety embedded by a sufficiently positive (pluri-) adjoint linear series are Du Bois but not rational in general. Moreover, the secant variety can be p-Du Bois, in the sense of Shen-Venkatesh-Vo, if certain Hodge numbers are zero; however it is almost never pre-1-rational. So unlike local complete intersections, p-Du Bois does not imply (p-1)-rational for secant varieties. The talk is based on the joint work with C-C. Chou and with S. Olano and D. Raychaudhury.