辛双有理几何的若干研究进展  

Some research progress on symplectic birational geometry

作　　者：胡建勋 李天军 阮勇斌 章唯一 Jianxun Hu;Tian-Jun Li;Yongbin Ruan;Weiyi Zhang

机构地区：[1]中山大学数学学院,广州510275 [2]School of Mathematics,University of Minnesota,Minneapolis,MN 55455,USA [3]浙江大学数学高等研究院,杭州310058 [4]Mathematics Institute,University of Warwick,Coventry,CV47AL,UK

出　　处：《中国科学：数学》2024年第12期2099-2120,共22页Scientia Sinica：Mathematica

基　　金：国家重点研发计划(批准号:2023YFA100980001)资助项目。

摘　　要：20世纪80年代,日本数学家Mori提出了极小模型纲领,其核心思想之一是把代数簇二分为单直纹簇和非单直纹簇来研究.对单直纹簇来说,试图理解以Fano簇为纤维的纤维化结构;而对于非单直纹簇,则要寻找它的极小模型.辛双有理几何是研究辛流形的双有理等价分类,并尝试将Mori的分类理论拓展到辛几何领域.本文综述该领域的最新进展,内容包括双有理配边等价、辛单直纹和有理连通辛流形、4维辛流形的Kodaira维数及近复流形的双有理几何等.In the 1980s,the Japanese mathematician Mori established his minimal model program.The point of the program is to classify algebraic varieties into two classes:uniruled and non-uniruled varieties.For the uniruled varieties,one tries to understand the structure of fibrations with Fano fibers,and for the non-uniruled varieties,one hopes to find their minimal models.Symplectic birational geometry is a field to study the classification of symplectic manifolds under symplectic birational equivalence,and to extend Mori’s minimal model program to symplectic category.In this paper,we survey some recent progress on symplectic birational geometry,including birational cobordism,uniruled and rationally connected symplectic manifolds,Kodaira dimension of symplectic 4-manifolds,and birational geometry of almost complex manifolds.

关 键 词：双有理配边 单直纹辛流形 有理连通辛流形 Gromov-Witten不变量 Kodaira维数 近复流形 

分 类 号：O186[理学—数学]