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

doi:10.3969/j.issn.1000-5641.2015.01.010

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

doi: 10.3969/j.issn.1000-5641.2015.01.010

林洁珠¹,
叶轩明^2, ,

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

基金项目:

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)

详细信息

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.

通讯作者:
叶轩明, 男, 讲师,研究方向为复几何、复代数几何.

中图分类号: O186
计量
- 文章访问数: 1206
- HTML全文浏览量: 17
- PDF下载量: 1187
- 被引次数: 0
出版历程
- 收稿日期: 2014-03-01
- 刊出日期: 2015-01-25

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-Ming^{2
, ,}

1.
School of Mathematics And Information Science, Guangzhou University, Key Laboratory of Mathematics, and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou 510006, China;

摘要

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 Lp,q的上同调群.在文献[2]中, 作者通过研究X各阶形变中与 Lp,q拟同构的层链 Bp,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 Lp,q上同调计算的解析证明.
- 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 Lp,q, and represented Bott-Chern cohomology groups or Aeppli cohomology groups as the cohomology groups of Lp,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.
- Bott-Chern cohomology /
- Aeppli cohomology /
- deformation /
- obstruction /
- kodaira spencer class

HTML全文

参考文献(1)

[1]

SCHWEITZER M. Autour de la cohomologie de Bott-Chern [J/OL]. arXiv:0709 3528v1, 2007[2014-03-06].http://arxiv.org/abs/0709.3528.
LIN J Z, YE X M. The jumping phenomenon of the dimensions of Bott-Chern cohomology groups and Aeppli cohomology groups [J/OL]. arXiv: 1403 0285v2, 2014[2014-03-06].http://arxiv.org/abs/1403.0285.
KODAIRA K. Complex manifolds and deformation of complex structures [M]. New York: Springer, 1986.
VOISIN C. Hodge theory and complex algebraic geometry I [M]. London: Cambridge University Press, 2002.
ANGELLA D. The cohomologies of the Iwasawa manifold and of its small deformations [J]. J Geom Anal, 2013, 23(3): 1355-1378.
YE X M. The jumping phenomenon of Hodge numbers [J]. Pacific Journal of Mathematics, 2008, 235(2): 379-398.
YE X M. The jumping phenomenon of the dimensions of cohomology groups of tangent sheaf [J]. Acta Mathematica Scientia, 2010, 30(5):

1746-1758.
VOISIN C. Symétrie miroir [M]. Paris: Société Mathématique de France, 1996.
FRÖLICHER A. Relations between the cohomology groups of Dolbeault and topological invariants [J], Proc Nat Acad Sci USA, 1955: 641-644.
BOTT R, CHERN S -S. Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections [J], Acta Math, 1965: 71-112.
AEPPLI A. On the cohomology structure of Stein manifolds [J], Proc Conf Complex Analysis (Minneapolis, Minn., 1964), 1965: 58-70.

施引文献

资源附件(0)

访问统计