华东师范大学学报(自然科学版) ›› 2015, Vol. 2015 ›› Issue (1): 84-94.doi: 10.3969/j.issn.1000-5641.2015.01.010

• 应用数学与基础数学 • 上一篇    下一篇

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1,叶轩明2   

  1. 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
  • 收稿日期:2014-03-01 出版日期:2015-01-25 发布日期:2015-03-29
  • 通讯作者: 叶轩明, 男, 讲师,研究方向为复几何、复代数几何. E-mail:yexm3@mail.sysu.edu.cn
  • 作者简介: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.
  • 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-Zhu1, YE  Xuan-Ming2   

  1. 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:2014-03-01 Online:2015-01-25 Published:2015-03-29

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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.

关键词: Bott-Chern上同调群, Aeppli上同调群, 复结构形变, 障碍,  Kodaira Spencer类

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.

Key words: Bott-Chern cohomology, Aeppli cohomology, deformation, obstruction, kodaira spencer class

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