A -Poincaré lemma for forms near an isolated complex singularity

Authors:
Adam Harris and Yoshihiro Tonegawa

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

Published electronically:
February 20, 2003

MathSciNet review:
1990620

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be an analytic subvariety of complex Euclidean space with isolated singularity at the origin, and let be a smooth form of type defined on . The main result of this note is a criterion for solubility of the equation . This implies a criterion for triviality of a Hermitian- holomorphic line bundle in a neighbourhood of the origin.

**[1]**Akahori, T., Miyajima, K.*An analogy of Tian-Todorov theorem on deformations of**-structures*, Compositio Math. 85 (1993) 57-85 MR**94b:32028****[2]**Bando, S.,*Removable singularities for holomorphic vector bundles*, Tohoku Math. J. 43 (1991) 61-67 MR**92a:32037****[3]**Dimca, A.,*Singularities and topology of hypersurfaces*, Springer Universitext 1992 MR**94b:32058****[4]**Federer, H.,*Geometric measure theory*, Grund. Math. Wiss. 153, Springer Berlin, Heidelberg, New York 1969 MR**41:1976****[5]**Griffiths, P., and Harris, J.,*Principles of Algebraic Geometry*, Wiley 1978 MR**95d:14001****[6]**Harris, A., and Tonegawa, Y.*-curvature and the Cauchy-Riemann equation near an isolated singular point*, Nagoya Math. J. 164 (2001) 35-51**[7]**Hermann, R.,*Differential geometry and the calculus of variations*, Math. Sci. Eng. 49, Academic Press 1968 MR**38:1635****[8]**Milnor, J.*Singular points of complex hypersurfaces*, Ann. Math. Stud. 61, Princeton 1968 MR**39:969****[9]**Miyajima, K.*A note on Bogomolov-type smoothness on deformations of regular parts of isolated singularities*, Proc. Am. Math. Soc. 125, No. 2 (1997) 485-492 MR**97d:32027****[10]**Weinstein, A.,*Symplectic manifolds and their Lagrangian submanifolds*, Adv. Math. 6 (1971) 329-346 MR**44:3351**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
14J17,
32B15,
32S05,
32W05

Retrieve articles in all journals with MSC (2000): 14J17, 32B15, 32S05, 32W05

Additional 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

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

DOI:
https://doi.org/10.1090/S0002-9939-03-06875-8

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

Article copyright:
© Copyright 2003
American Mathematical Society