A {̄∂}∂–Poincaré lemma for forms near an isolated complex singularity

Harris, Adam; Tonegawa, Yoshihiro

doi:10.1090/S0002-9939-03-06875-8

A $\bar \{\partial \}\partial$–Poincaré lemma for forms near an isolated complex singularity
HTML articles powered by AMS MathViewer

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

PDF | Request permission

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

Takao Akahori and Kimio Miyajima, An analogy of Tian-Todorov theorem on deformations of CR-structures, Compositio Math. 85 (1993), no. 1, 57–85. MR 1199204
Shigetoshi Bando, Removable singularities for holomorphic vector bundles, Tohoku Math. J. (2) 43 (1991), no. 1, 61–67. MR 1088714, DOI 10.2748/tmj/1178227535
Alexandru Dimca, Singularities and topology of hypersurfaces, Universitext, Springer-Verlag, New York, 1992. MR 1194180, DOI 10.1007/978-1-4612-4404-2
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original. MR 1288523, DOI 10.1002/9781118032527
Harris, A., and Tonegawa, Y. $L^{p}$–curvature and the Cauchy–Riemann equation near an isolated singular point, Nagoya Math. J. 164 (2001) 35–51
Robert Hermann, Differential geometry and the calculus of variations, Mathematics in Science and Engineering, Vol. 49, Academic Press, New York-London, 1968. MR 0233313
John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612
Kimio Miyajima, A note on the Bogomolov-type smoothness on deformations of the regular parts of isolated singularities, Proc. Amer. Math. Soc. 125 (1997), no. 2, 485–492. MR 1363432, DOI 10.1090/S0002-9939-97-03712-X
Alan Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math. 6 (1971), 329–346 (1971). MR 286137, DOI 10.1016/0001-8708(71)90020-X