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

1979; American Mathematical Society; Volume: 252; Linguagem: Inglês

10.1090/s0002-9947-1979-0534116-x

1088-6850

Hans Rudi Fischer, Floyd L. Williams,

Geometry and complex manifolds

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.