Article

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
Expand
  • 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

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.

Cite this article

LIN Jie-Zhu , YE Xuan-Ming . An analytic proof for the formula of the first order obstruction making the dimensions of Bott-Chern cohomology groups and Aeppli cohomology groups jumping[J]. Journal of East China Normal University(Natural Science), 2015 , 2015(1) : 84 -94 . DOI: 10.3969/j.issn.1000-5641.2015.01.010

References

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.
Outlines

/