Nagoya Mathematical Journal

{$L\sp p$}-curvature and the Cauchy-Riemann equation near an isolated singular point

Adam Harris and Yoshihiro Tonegawa

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Let $X$ be a complex $n$-dimensional reduced analytic space with isolated singular point $x_{0}$, and with a strongly plurisubharmonic function $\rho:X \to [0,\infty)$ such that $\rho(x_{0}) = 0$. A smooth Kähler form on $X\setminus\{x_{0}\}$ is then defined by ${\bf i}\partial\bar{\partial}\rho$. The associated metric is assumed to have $L^{n}_{\rm{loc}}$-curvature, to admit the Sobolev inequality and to have suitable volume growth near $x_{0}$. Let $E \to X\setminus\{x_{0}\}$ be a Hermitian-holomorphic vector bundle, and $\xi$ a smooth $(0,1)$-form with coefficients in $E$. The main result of this article states that if $\xi$ and the curvature of $E$ are both $L^{n}_{\rm{loc}}$, then the equation $\bar{\partial}u = \xi$ has a smooth solution on a punctured neighbourhood of $x_{0}$. Applications of this theorem to problems of holomorphic extension, and in particular a result of Kohn-Rossi type for sections over a $CR$-hypersurface, are discussed in the final section.

Article information

Nagoya Math. J., Volume 164 (2001), 35-51.

First available in Project Euclid: 27 April 2005

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Zentralblatt MATH identifier

Primary: 32D20: Removable singularities
Secondary: 32L10: Sheaves and cohomology of sections of holomorphic vector bundles, general results [See also 14F05, 18F20, 55N30] 32W10: $\overline\partial_b$ and $\overline\partial_b$-Neumann operators


Harris, Adam; Tonegawa, Yoshihiro. {$L\sp p$}-curvature and the Cauchy-Riemann equation near an isolated singular point. Nagoya Math. J. 164 (2001), 35--51.

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  • A. Andreotti and E. Vesentini, Carleman estimates for the Laplace-Beltrami equation on complex manifolds , Publ. Math. I.H.E.S., 25 (1965), 81–130.
  • S. Bando, Removable singularities for holomorphic vector bundles , Tohoku Math. J., 43 (1991), 61–67.
  • S. Bando, S. Kasue and H. Nakajima, On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth , Invent. Math., 97 (1989), 313–349.
  • N.P. Buchdahl and A. Harris, Holomorphic connections and extension of complex vector bundles , Math. Nachr., 204 (1999), 29–39.
  • A. Dimca, Singularities and topology of hypersurfaces, Springer Universitext (1992).
  • H. Federer, Geometric measure theory, Grund. Math. Wiss. Bd 153, Springer Berlin, Heidelberg, New York (1969).
  • R.M. Goresky and R.P. MacPherson, Intersection homology theory , Topology, 19 (1980), 135–162.
  • H. Grauert, Characterisierung der Holomorphiegebiete durch die vollstäntige Kählerische Metrik , Math. Ann., 131 (1956), 38–75.
  • A. Harris and Y. Tonegawa, Analytic continuation of vector bundles with $L^p$- curvature , Int. J. Math., 11, No.1 (2000), 29–40.
  • L. Hömander, $L^2$- estimates and existence theorems for the $\bar\partial$-operator , Acta Math., 113 (1965), 89–152.
  • J. Kohn and G. Folland, The Neumann problem for the Cauchy-Riemann complex, Ann. Math. Stud. 75, Princeton (1972).
  • J. Kohn and H. Rossi, On the extension of holomorphic functions from the boundary of a complex manifold , Ann. Math., 81 (1965), 451–472.
  • H.B. Laufer, On $\Bbb C\Bbb P_1$ as an exceptional set , in “Recent developments in several complex variables” (J.E. Fornaess, Ed.) Ann. Math. Stud., Princeton 1981.
  • T. Ohsawa, Cheeger-Goresky-MacPherson's conjecture for varieties with isolated singularities , Math. Z., 206 (1991), 219–224.
  • ––––, On the $L^2$-cohomology groups of isolated singularities , Adv. Stud. Pure Math., 22 (1993), 247–263.
  • L. Saper, $L^2$- cohomology of Kähler varieties with isolated singularities , J. Differential Geom., 36, No.1 (1992), 89–161.