应用数学与基础数学

1阶复结构形变中产生Bott-Chern上同调群和Aeppli上同调群维数跳跃的障碍公式的解析证明

  • 林洁珠 ,
  • 叶轩明
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  • 1. 广州大学 数学与信息科学学院, 数学与交叉学科 广东普通高校重点实验室, 广州 510006 2. 中山大学 数学与计算科学学院数学系, 广州 510275
第一作者: 林洁珠, 女, 副教授,研究方向为数学物理、复微分几何. E-mail: jlin@gzhu.edu.cn.

收稿日期: 2014-03-01

  网络出版日期: 2015-03-29

基金资助

国家青年基金(11201090, 11201491);

博士点新教师类项目(20124410120001,201201711)

高校基本科研业务费青年教师培育项目(34000-3161248)

An analytic proof for the formula of the first order obstruction making the dimensions of Bott-Chern cohomology groups and Aeppli cohomology groups jumping

  • LIN Jie-Zhu ,
  • YE Xuan-Ming
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  • 1. School of Mathematics And Information Science, Guangzhou University, Key Laboratory of Mathematics, and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou 510006, China;
    2. School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275, China

Received date: 2014-03-01

  Online published: 2015-03-29

摘要

设X为一个紧致复流形,考虑\,$X$\,的任一复结构形变族 : X ! B ,则X的Bott-Chern上同调群和Aeppli上同调群的维数在此变化过程中可能产生跳跃现象. 在文献[1]中Schweitzer将Bott-Chern上同调群和Aeppli上同调群表示成为某一个层链 L•p,q的上同调群.在文献[2]中, 作者通过研究X各阶形变中与 L•p,q拟同构的层
链 B•p,q的超上同调群等价类元素在延拓过程中的 障碍来研究这一跳跃现象,得到了产生此障碍的公式. 本文将给出1阶障碍公式的另一个用 L•p,q上同调计算的解析证明.

本文引用格式

林洁珠 , 叶轩明 . 1阶复结构形变中产生Bott-Chern上同调群和Aeppli上同调群维数跳跃的障碍公式的解析证明[J]. 华东师范大学学报(自然科学版), 2015 , 2015(1) : 84 -94 . DOI: 10.3969/j.issn.1000-5641.2015.01.010

Abstract

Let X be a compact complex manifold, and let  : X ! B be a small deformation of X, the dimensions of the Bott-Chern cohomology groups or Aeppli
cohomology groups may vary under this deformation. In [1], M. Schweitzer constructed a complex of sheaves L•p,q, and represented Bott-Chern cohomology groups or Aeppli cohomology groups as the cohomology groups of L•p,q. In [2], the author have studied this jumping phenomenon by studying the deformation obstructions of a hypercohomology class of a complex of sheaves B• p,q which is quasi-isomorphic to L• p,q[1]. In particular, they obtain an explicit formula for the obstructions. In this paper, the formula of the first order obstruction is proved in another way by using cohomology of L• p,q.

参考文献

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