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$ \overline{\partial}$-equation on $ (p,q)$-forms on conic neighbourhoods of $ 1$-convex manifolds


Author: Jasna Prezelj
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 32W05, 32E05; Secondary 32E10, 32C15, 32C35, 32W05
DOI: https://doi.org/10.1090/proc/13603
Published electronically: May 24, 2017
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Abstract: Let $ X$ be a $ 1$-convex manifold with the exceptional set $ S$, which is also a manifold, $ Z$ a complex manifold, $ Z \rightarrow X$ a holomorphic submersion, $ a: X \rightarrow Z$ a holomorphic section and $ S \subset U \Subset X$ an open relatively compact $ 1$-convex set. We construct a metric on a vector bundle $ E \rightarrow Z$ restricted to a neighbourhood $ V$ of $ a(U),$ conic along $ a(S)$ with at most polynomial poles at $ a(S)$ and positive Nakano curvature tensor in bidegree $ (p,q).$


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Additional Information

Jasna Prezelj
Affiliation: Faculty of Mathematics, Natural Sciences and Information Technologies, University of Primorska, Glagoljaška 8, SI-6000 Koper, Slovenia; Faculty of Mathematics and Physics, Department of Mathematics, University of Ljubljana, Jadranska 21, SI-1000 Ljubljana, Slovenia
Email: jasna.prezelj@fmf.uni-lj.si

DOI: https://doi.org/10.1090/proc/13603
Keywords: 1-convex set, Nakano positive metric, K\"ahler form, curvature tensor
Received by editor(s): July 4, 2016
Received by editor(s) in revised form: November 10, 2016
Published electronically: May 24, 2017
Additional Notes: The author was supported by Slovenian Research Agency research program P1-0291 and research project J1-5432. Part of the paper was written while the author was visiting NTNU, Trondheim, Norway, and she wishes to thank this institution for its hospitality. Particular thanks go to the reviewer for his careful reading of the article.
Communicated by: Franc Forstneric
Article copyright: © Copyright 2017 American Mathematical Society