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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Global residues and intersections on a complex manifold

Author: James R. King
Journal: Trans. Amer. Math. Soc. 192 (1974), 163-199
MSC: Primary 32C30
MathSciNet review: 0338433
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Abstract: This paper is the study of a class of forms $ \eta $ on a complex manifold V which are smooth on $ V - W$ and have poles of kernel type on a complex submanifold W of codimension d; such a form is one whose pull-back to the monoidal transform of V along W has a logarithmic pole. A global existence theorem is proved which asserts that any smooth form $ \varphi $ on W of filtration s (no (p, q) components with $ p < s$) is the residue of a form $ \eta $ of filtration $ s + d$ such that $ d\eta $ is smooth on V. This result is used to construct global kernels for $ \bar \partial $ which establish similar global existence theorems for W with singularities. We then establish formulas connecting intersection and wedge product on the d-cohomology theory of Dolbeault which preserve the Hodge filtration. A number of results are also proved on the integrability of $ {f^\ast}\eta $ where f is a rather general holomorphic map.

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Keywords: Residue forms, intersection of analytic sets, Dolbeault cohomology, kernels on complex manifolds
Article copyright: © Copyright 1974 American Mathematical Society

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