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The Bergman kernel on forms: General theory


Author: Andrew Raich
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 32A25, 32A55, 32W05
DOI: https://doi.org/10.1090/proc/13921
Published electronically: August 14, 2018
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Abstract: The goal of this paper is to explore the Bergman projection on forms. In particular, we show that some of most basic facts used to construct the Bergman kernel on functions, such as pointwise evaluation in $ L^2_{0,q}(\Omega )\cap \ker \bar \partial _q$, fail for $ (p,q)$-forms, $ q \geq 1$, $ p\geq 0$. We do, however, provide a careful construction of the Bergman kernel and explicitly compute the Bergman kernel on $ (0,n-1)$-forms. For the ball in $ \mathbb{C}^2$, we also show that the size of the Bergman kernel on $ (0,1)$-forms is not governed by the control metric, in stark contrast to the Bergman kernel on functions.


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Additional Information

Andrew Raich
Affiliation: Department of Mathematical Sciences, SCEN 309, University of Arkansas, Fayetteville, Arkansas 72701
Email: araich@uark.edu

DOI: https://doi.org/10.1090/proc/13921
Keywords: Bergman projection, Bergman kernel
Received by editor(s): June 2, 2017
Received by editor(s) in revised form: July 20, 2017
Published electronically: August 14, 2018
Additional Notes: The author was partially supported by NSF grant DMS-1405100.
Communicated by: Harold P. Boas
Article copyright: © Copyright 2018 American Mathematical Society

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