Riv. Mat. Univ. Parma, Vol. 13, No. 2, 2022

Tatsuo Suwa [a]

Relative Dolbeault cohomology

Pages: 307-352
Received: 24 November 2021
Accepted: 10 March 2022
Mathematics Subject Classification: 14B15, 14F08, 32A45, 32C35, 32C37, 35A27, 46A20, 46F15, 46M20, 55N05, 58J15.
Keywords: Dolbeault cohomology of an open embedding, Cech-Dolbeault cohomology, relative Dolbeault theorem, complex analytic Alexander morphism, Sato hyperfunctions.
Author address:
[a]: Hokkaido University, Department of Mathematics, Sapporo, Japan

This work was supported by JSPS Grants-in-Aid for Scientific Research Grant Numbers JP24540060, JP16K05116, JP20K03572

In memory of Pierre Dolbeault

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Abstract: This is a partially expository paper, in which the notion of relative Dolbeault cohomology and related topics are reviewed and discussed together with various examples and applications. We deal with this cohomology theory from two viewpoints. One is the Cech theoretical approach, which is convenient to define such operations as the cup product and integration and leads to the study of local duality. Along the way we also establish some notable canonical isomorphisms among various cohomologies. The other is to regard it as the cohomology of a certain complex, which is interpreted as a notion dual to the mapping cone in the theory of derived categories. This approach shows that the cohomology goes well with derived functors. In any case the relative Dolbeault cohomology turns out to be canonically isomorphic with the local (relative) cohomology of A. Grothendieck and M. Sato with coefficients in the sheaf of holomorphic forms so that it provides a handy way of representing the latter. We also give some examples and indicate applications, including simple explicit expressions of Sato hyperfunctions, fundamental operations on them and related local duality theorems.

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