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The Dolbeault complex in infinite dimensions I

Author: László Lempert
Journal: J. Amer. Math. Soc. 11 (1998), 485-520
MSC (1991): Primary 32C10, 32L20, 58B12
Part II: J. Amer. Math. Soc. (1999), 775-793
MathSciNet review: 1603858
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Abstract: In this paper we introduce certain basic notions concerning infinite dimensional complex manifolds, and prove that the Dolbeault cohomology groups of infinite dimensional projective spaces, with values in finite rank vector bundles, vanish. Some applications of such vanishing theorems are discussed; e.g., we classify vector bundles of finite rank over infinite dimensional projective spaces. Finally, we prove a sharp theorem on solving the inhomogeneous Cauchy–Riemann equations on affine spaces.

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Additional Information

László Lempert
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
MR Author ID: 112435

Received by editor(s): February 4, 1997
Additional Notes: This research was partially supported by an NSF grant, and also by the Mathematical Sciences Research Institute, Berkeley.
Article copyright: © Copyright 1998 American Mathematical Society