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Kohn decomposition for forms on coverings of complex manifolds constrained along fibres


Authors: A. Brudnyi and D. Kinzebulatov
Journal: Trans. Amer. Math. Soc. 369 (2017), 167-186
MSC (2010): Primary 32A38, 32K99
DOI: https://doi.org/10.1090/tran/6633
Published electronically: February 12, 2016
MathSciNet review: 3557771
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Abstract: The classical result of J.J. Kohn asserts that over a relatively compact subdomain $ D$ with $ C^\infty $ boundary of a Hermitian manifold whose Levi form has at least $ n-q$ positive eigenvalues or at least $ q+1$ negative eigenvalues at each boundary point, there are natural isomorphisms between the $ (p,q)$ Dolbeault cohomology groups defined by means of $ C^\infty $ up to the boundary differential forms on $ D$ and the (finite-dimensional) spaces of harmonic $ (p,q)$-forms on $ D$ determined by the corresponding complex Laplace operator. In the present paper, using Kohn's technique, we give a similar description of the $ (p,q)$ Dolbeault cohomology groups of spaces of differential forms taking values in certain (possibly infinite-dimensional) holomorphic Banach vector bundles on $ D$. We apply this result to compute the $ (p,q)$ Dolbeault cohomology groups of some regular coverings of $ D$ defined by means of $ C^\infty $ forms constrained along fibres of the coverings.


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Additional Information

A. Brudnyi
Affiliation: Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta T2N 1N4, Canada
Email: abrudnyi@ucalgary.ca

D. Kinzebulatov
Affiliation: The Fields Institute, 222 College Street, Toronto, Ontario M5T 3J1, Canada
Email: dkinzebu@fields.utoronto.ca

DOI: https://doi.org/10.1090/tran/6633
Keywords: Kohn decomposition, holomorphic Banach vector bundle, harmonic form
Received by editor(s): March 5, 2014
Received by editor(s) in revised form: November 4, 2014, and December 16, 2014
Published electronically: February 12, 2016
Additional Notes: The authors’ research was partially supported by NSERC
Article copyright: © Copyright 2016 American Mathematical Society