刘 杰

The tropicalization process assigns to an algbraic variety a polyhedral complex with extra structure that records certain degeneration data. In this talk, I will introduce the tropicalization of a family of algebraic curves and explain the connection between the geometry of the algebro-geometric side and the tropical side. I will then discuss a few applications of this construction to the Severi problem. This is joint work with Karl Christ and Ilya Tyomkin.

An automorphism of a smooth complex projective variety \(X\) is called numerically trivial if it induces the trivial action on the cohomology groups with rational coefficients. We denote by \(Aut_\mathbb{Q}(X)\) the group of such automorphisms. For algebraic curves, \(Aut_\mathbb{Q}(X)\) coincides with the identity component of \(Aut(X)\). The situation is more complicated for algebraic surfaces. Nevertheless, Jin-Xing Cai proved that \(\left|Aut_\mathbb{Q}\left(X\right)\right|\le 4\) if \(X\) is a surface of general type with \(\chi\left(\mathcal{O}_X\right)\geq 189\). Irregular minimal surfaces of general type with \(\left|Aut_\mathbb{Q}\left(X\right)\right|=4\) turns out to be isogenous to a product of curves by a previous joint work with Jin-Xing Cai. In this talk, I will present some examples of regular surfaces of general type with \(\left|Aut_\mathbb{Q}\left(X\right)\right|=4\), as well as some work in progress towards a classification similar to the irregular case. This is a joint work with Professor Jin-Xing Cai.

I will report a joint work in progress with C. Li and R. Xiong. The Calogero-Moser (CM) system, which is also called the hypergeometric system, is a generalization of the Gauss hypergeometric equation. We give explicit formulae for the integrals of motions of this system. Moreover, we give a presentation for the quantum cohomolgy ring of the Springer resolution with relations given by the integrals of motions of the CM system. 

I will describe a technique to lift algebraic equivalence of one cycles on smooth projective varieties to deformation equivalence between stable maps, and some of its applications in geometry and arithmetic. This is joint work with János Kollár.