Twisted homology and cohomology groups associated to the Wirtinger integral. (English) Zbl 1188.14014

Watanabe, Humihiko

J. Math. Soc. Japan 59, No. 4, 1067-1080 (2007).

Summary: The first half of this paper (Section 1) deals with the structure of the twisted homology group associated to the Wirtinger integral. A basis of the first homology group is given, and the vanishing of the homology groups of the other dimensions is proved. The second half (Section 2) deals with the structure of the twisted cohomology groups associated to the Wirtinger integral. The isomorphism between the twisted cohomology groups and the cohomology groups associated to a certain subcomplex of the de Rham complex is established, and a basis of the first cohomology group of this subcomplex (therefore, of the first twisted cohomology group) is given. The vanishing of the cohomology groups of the other dimensions is also proved.

MSC:

14F40	de Rham cohomology and algebraic geometry
14K25	Theta functions and abelian varieties
32S40	Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
33E05	Elliptic functions and integrals
33C05	Classical hypergeometric functions, \({}_2F_1\)

Keywords:

Wirtinger integral; theta function; twisted (co)homology group; de Rham cohomology