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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Semiglobal results for $ \overline\partial$ on complex spaces with arbitrary singularities, Part II


Authors: Nils Øvrelid and Sophia Vassiliadou
Journal: Trans. Amer. Math. Soc. 363 (2011), 6177-6196
MSC (2010): Primary 32B10, 32J25, 32W05
Published electronically: July 14, 2011
MathSciNet review: 2833549
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Abstract: We obtain some $ L^2$-results for $ \overline\partial$ on forms that vanish to high order on the singular set of a complex space. As a consequence of our main theorem we obtain weighted $ L^2$-solvability results for compactly supported $ \overline\partial$-closed $ (p,q)$-forms $ (0\le p\le n, 1\le q< n)$ on relatively compact subdomains $ \Omega$ of the complex space that satisfy $ H^{n-q}(\Omega, \mathcal{S})=0=H^{n-q+1}(\Omega, \mathcal{S})$ for every coherent $ \mathcal{O}_X$-module $ \mathcal{S}$. The latter result can be used to give an alternate proof of a theorem of Merker and Porten.


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Additional Information

Nils Øvrelid
Affiliation: Department of Mathematics, University of Oslo, P.B. 1053 Blindern, Oslo, N-0316 Norway
Email: nilsov@math.uio.no

Sophia Vassiliadou
Affiliation: Department of Mathematics, Georgetown University, Washington, DC 20057
Email: sv46@georgetown.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05274-8
PII: S 0002-9947(2011)05274-8
Keywords: Cauchy-Riemann equation, singularity, cohomology groups
Received by editor(s): June 19, 2009
Published electronically: July 14, 2011
Additional Notes: The research of the second author was partially supported by NSF grant DMS-0712795
Article copyright: © Copyright 2011 American Mathematical Society