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Transactions of the American Mathematical Society

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Solving $ {\overline{\partial}}$ with prescribed support on Hartogs triangles in $ {\mathbb{C}}^2$ and $ {\mathbb{C}}\mathbb{P}^2$


Authors: Christine Laurent-Thiébaut and Mei-Chi Shaw
Journal: Trans. Amer. Math. Soc. 371 (2019), 6531-6546
MSC (2010): Primary 32C35, 32W05; Secondary 32C37
DOI: https://doi.org/10.1090/tran/7545
Published electronically: September 25, 2018
MathSciNet review: 3937336
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Abstract: In this paper, we consider the problem of solving the Cauchy-Riemann equation with prescribed support in a domain of a complex manifold for forms or currents. We are especially interested in the case when the domain is a Hartogs triangle in $ \mathbb{C}^2$ or $ \mathbb{C}\mathbb{P}^2$. In particular, we show that the strong $ L^2$ Dolbeault cohomology group on the Hartogs triangle in $ \mathbb{C}\mathbb{P}^2$ is infinitely dimensional.


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Additional Information

Christine Laurent-Thiébaut
Affiliation: Université Grenoble-Alpes, Institut Fourier, Grenoble, F-38041, France; and CNRS UMR 5582, Institut Fourier, Saint-Martin d’Hères F-38402, France

Mei-Chi Shaw
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

DOI: https://doi.org/10.1090/tran/7545
Received by editor(s): September 14, 2016
Received by editor(s) in revised form: January 15, 2018
Published electronically: September 25, 2018
Additional Notes: Both authors were partially supported by a grant from the AGIR program of Grenoble INP and Université Grenoble-Alpes, awarded to the first author.
The second author is partially supported by an NSF grant.
Article copyright: © Copyright 2018 American Mathematical Society