高维代数流形Moduli空间和纤维丛的几何及其正特征代数簇相关问题

项目名称： 高维代数流形Moduli空间和纤维丛的几何及其正特征代数簇相关问题

项目编号： No.11271070

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 张毅

作者单位： 复旦大学

项目金额： 50万元

中文摘要： 题目：高维代数流形Moduli空间和纤维丛的几何及其正特征代数簇相关问题。 本项目拟围绕着高维代数流形Moduli空间紧化（compactification），Moduli空间上的上同调研究，以及纤维丛的形变（deformation）分类开展工作。 具体是以下四个方面： I. Calabi-Yau类流形Moduli空间（包括局部对称算术簇）的整体几何性质和代数几何紧化（compactification）的几何构造精细化。 II. Calabi-Yau类流形Moduli（包括局部对称算术簇）紧化空间上local system 向量丛的各类上同调； III. Calabi-Yau类流形Moduli空间的子簇和Calabi-Yau 流形纤维丛几何结构这些方向的研究。 IV. 正特征代数簇提升性质及其在Calabi-Yau类流形相关问题上的应用

中文关键词： 模空间；Hodge 理论；紧化；阿贝尔簇及卡拉比-丘流形；正特征代 数簇的提升性

英文摘要： Title: The Geometry of Moduli Spaces and families of of Higher dimensional algebraic manifolds,including related topics on varities with positive characteristic. We plan to study compactifications of moduli spaces of higher dimensional algebraic manifolds, the cohomology groups of moduli spaces and the deformations of families of algebraic manifolds. Our program will contains the following four topics: I. The Global geometry of moduli spaces of polarized Calabi-Yau-like manifolds(including the geometry of locally symmetric arithmetic varieties) and the explicit compactifications of moduli spaces. II. Cohomology groups of local systems on moduli spaces of polarized Calabi-Yau-like manifolds， including cohomology groups of locally symmetric varieties. III. Sub-varieties of moduli spaces of polarized Calabi-Yau-like manifolds and the fiberation structures of Calabi-Yau-like manifolds. IV.The liftable property of algebraic varieties with positive characteristic p, and its applications in problems related to Calabi-Yau-like manifolds. We will study our program from view of modern Hodge Theory.

英文关键词： Moduli space；Hodge Theory；compactification；Abelian variety and Calabi-Yau manifold；the liftable property of algebraic varieties with