Riv. Mat. Univ. Parma, Vol. 13, No. 2, 2022
Tatsuo Suwa ^{[a]}
Relative Dolbeault cohomology
Pages: 307352
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, CechDolbeault 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 GrantsinAid for Scientific Research Grant Numbers JP24540060, JP16K05116, JP20K03572
In memory of Pierre Dolbeault
Full Text (PDF)
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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