刘 杰

The theory of complements is introduced by Shokurov when he proved the existence of flips for threefolds. It turns out that complements theory plays an important role in the study of higher dimensional geometry. In this talk, I will introduce some applications of this theory and survey some recent progress towards the boundedness of complements with DCC coefficients. Then we may turn to deal with pairs with [0,1] coefficients and prove the boundedness of complements for Calabi-Yau three-folds. This is a joint work with Jingjun Han and Qingyuan Xue.

Fano varieties and compact hyperkähler varieties are two important objects in the study of algebraic geometry. In this talk, we focus on two specific families of Fano fourfolds, cubic fourfolds and Gushel-Mukai fourfolds, and investigate their connection with hyperkähler varieties. We begin by recalling some basic notions and classical results of these two objects. Then we mainly study the interplay between them from the perspective of the K3 nature of the derived categories. This is based on the joint work with Zhiyu Liu and Shizhuo Zhang. Finally, motivated by a recently developed notion called atomic Lagrangian by Markman and Beckmann, I would like to briefly recall Bottini’s work of constructing a birational model of OG10 by using a specific atomic Lagrangian and discuss some related questions. 

We explore the applications of Lorentzian polynomials to the fields of algebraic geometry, analytic geometry and convex geometry. In particular, we establish a series of intersection theoretic inequalities, which we call rKT property, with respect to m-positive classes and Schur classes. We also study its convexity variants -- the geometric inequalities for m-convex functions on the sphere and convex bodies. Along the exploration, we prove that any finite subset on the closure of the cone generated by m-positive classes can be endowed with a polymatroid structure by a canonical numerical-dimension type function, extending our previous result for nef classes; and we prove Alexandrov-Fenchel inequalities for valuations of Schur type. We also establish various analogs of sumset estimates (Plunnecke-Ruzsa inequalities) from additive combinatorics in our contexts. This is a joint work with Jian Xiao. 

Additive action on projective variety is a variety which is equivariant compactification of vector group. Baohua Fu and Jun-Muk Hwang introduce the concept of Euler-symmetric varieties for the systematic understanding of the Picard number one smooth equivariant compactifications of vector groups. Euler-symmetric projective varieties are nondegenerate projective varieties admitting many-actions of Euler type. And those varieties are unique determined by its fundamental forms or its associated symbol system, but the relation between the algebraic properties of a symbol system and the geometric properties of the associated Euler-symmetric projective variety is very intriguing. How to understanding this relation is the key question. In this talk, I will give the relation between the complete intersection and the rank of its associated symbol system.