Abstract
Let be a field of characteristic 0. Let X be a smooth complete intersection over
of dimension n – k in the projective space
for given positive integers n and k. When
Terasoma and Konno provided an explicit representative (in terms of differential forms) of a basis for the primitive middle-dimensional algebraic de Rham cohomology
Later Dimca constructed another explicit representative of a basis of
. Moreover, he proved that his representative gives the same cohomology class as the previous representative of Terasoma and Konno. The goal of this article is to examine the above two different approaches without assuming that
and provide a similar comparison result for any field
Dimca’s argument depends heavily on the condition
and our idea is to find appropriate Cech-de Rham complexes and spectral sequences corresponding to those two approaches, which work without restrictions on
2020 Mathematics Subject Classification:
Acknowledgements
Jeehoon Park thanks KIAS (Korea Institute for Advanced Study), where the part of work was done, for its hospitality. The authors thank the anonymous referee for useful comments to improve the article.
Notes
1 More precisely, the isomorphism is due to Griffiths in the hypersurface case [3], Terasoma in the equal degree complete intersection case [11], and Konno in the general case [8].
2 See [9, Section 1.4] for physical explanation how to understand S as an action functional of a 0-dimensional quantum field theory.
3 Here, and, for
we think of
as a formal expression.
4 This was already introduced in (1.1)
5 More precisely, Terasoma [11] provided such an isomorphism in the case and Konno [8] extended the result of Terasoma to the general case when dj’s are not equal. Also see [3] for the pioneering work of Griffiths in the case k = 1, hypersurface case.
6 This fact is crucially used to develop the deformation theory of period integrals of in [7].
7 For example,