## A $\bar \{\partial \}\partial$–Poincaré lemma for forms near an isolated complex singularity

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- by Adam Harris and Yoshihiro Tonegawa
- Proc. Amer. Math. Soc.
**131**(2003), 3329-3334 - DOI: https://doi.org/10.1090/S0002-9939-03-06875-8
- Published electronically: February 20, 2003
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## Abstract:

Let $X$ be an analytic subvariety of complex Euclidean space with isolated singularity at the origin, and let $\eta$ be a smooth form of type $(1.1)$ defined on $X \setminus \{0\}$. The main result of this note is a criterion for solubility of the equation $\bar {\partial }\partial u = \eta$. This implies a criterion for triviality of a Hermitian– holomorphic line bundle $(L,h)\to X\setminus \{0\}$ in a neighbourhood of the origin.## References

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## Bibliographic Information

**Adam Harris**- Affiliation: Department of Mathematics and Statistics, University of Melbourne, Parkville, Victoria 3010, Australia
- Address at time of publication: Department of Mathematics & Computer Science, University of New England, Armidale, New South Wales 2351, Australia
- MR Author ID: 607698
- Email: harris@ms.unimelb.edu.au
**Yoshihiro Tonegawa**- Affiliation: Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
- Email: tonegawa@math.sci.hokudai.ac.jp
- Received by editor(s): September 18, 2001
- Received by editor(s) in revised form: June 1, 2002
- Published electronically: February 20, 2003
- Communicated by: Mei-Chi Shaw
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**131**(2003), 3329-3334 - MSC (2000): Primary 14J17, 32B15, 32S05, 32W05
- DOI: https://doi.org/10.1090/S0002-9939-03-06875-8
- MathSciNet review: 1990620