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Transactions of the American Mathematical Society

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Complex-foliated structures. I. Cohomology of the Dolbeault-Kostant complexes

Authors: Hans R. Fischer and Floyd L. Williams
Journal: Trans. Amer. Math. Soc. 252 (1979), 163-195
MSC: Primary 58A30; Secondary 32L10, 58F06, 81C40
MathSciNet review: 534116
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Abstract: We study the cohomology of differential complexes, which we shall call Dolbeault-Kostant complexes, defined by certain integrable sub-bundles F of the complex tangent bundle of a manifold M. When M has a complex or symplectic structure and F is chosen to be the bundle of anti-holomorphic tangent vectors or, respectively, a ``polarization'' then the corresponding complexes are, respectively, the Dolbeault complex and (under further conditions) a complex introduced by Kostant in the context of geometric quantization. A simple condition on F insures that our complexes are elliptic. Assuming ellipticity and compactness of M, for example, one of our key results is a Hirzebruch-Riemann-Roch Theorem.

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Keywords: Sheaf cohomology, Chern class, Todd class, spectral sequence, elliptic complex
Article copyright: © Copyright 1979 American Mathematical Society