非线性代数群作用的若干问题

Non-linear algebraic groups are algebraic groups with non-trivial Abanese maps. This project is devoted to study the following question: let X be a variety, under what condition does X admit an action of a non-linear algebraic group? What geometric properties do these varieties have? We are planning to use techniques from Hodge theory and birational geometry to find existence of group actions on certain class of varieties, and apply them to solve related problems in algebraic geometry and complex geometry. For example, the well-known conjecture of Demailly, Peternell and Schneider concerning compact Kahler manifolds with nef anti-canonical divisors, Jordan property of the birational automorphism groups, and Shokurov's geometric characterization of toric varietis. We wish to bring new methods and results to the research of these questions from the viewpoint of actions of algebraic groups.

非线性代数群即具有非平凡Albanese映射的代数群。本项目致力于探讨如下问题：设X 是一个代数簇，在怎样的条件下X 容许有一个非线性代数群的作用？这些代数簇有什么几何性质？我们计划使用Hodge理论和双有理几何的技术在某些代数簇上寻找这种群作用的存在性，并将此项发现应用于解决代数几何与复几何中的相关问题，例如：Demailly，Peternell和Scheneider关于反典范除子是nef的紧Kahler流形的著名猜测，双有理自同构群的Jordan性质以及Shokurov关于环型簇的几何刻画等等。我们希望从代数群作用的角度给这些问题的研究带来新的方法和结果。