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\(L^p\)-curvature and the Cauchy-Riemann equation near an isolated singular point. (English) Zbl 1027.32027
Summary: 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 \({\mathbf i}\partial\overline\partial\rho\). The associated metric is assumed to have \(L^n_{\text{loc}}\)-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 \(L^n_{\text{loc}}\), then the equation \(\overline\partial u= \xi\) 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.

32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
32D20 Removable singularities in several complex variables
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
32D15 Continuation of analytic objects in several complex variables
Full Text: DOI
[1] Int. J. Math. 11 pp 29– (2000)
[2] DOI: 10.1007/BF02391775 · Zbl 0158.11002
[3] DOI: 10.1016/0040-9383(80)90003-8 · Zbl 0448.55004
[4] Grund. Math. Wiss. Bd 153 (1969)
[5] Singularities and topology of hypersurfaces (1992)
[6] DOI: 10.1002/mana.19992040103 · Zbl 0939.32018
[7] J. Differential Geom. 36 pp 89– (1992) · Zbl 0780.14010
[8] DOI: 10.1007/BF01389045 · Zbl 0682.53045
[9] Adv. Stud. Pure Math. 22 pp 247– (1993)
[10] DOI: 10.2748/tmj/1178227535 · Zbl 0736.32011
[11] DOI: 10.1007/BF02571337 · Zbl 0728.14022
[12] DOI: 10.1007/BF02684398 · Zbl 0138.06604
[13] in ”Recent developments in several complex variables” (J.E. Fornaess, Ed.) Ann. Math. Stud. (1981)
[14] DOI: 10.2307/1970624 · Zbl 0166.33802
[15] Ann. Math. Stud 75 (1972)
[16] DOI: 10.1007/BF01354665 · Zbl 0073.30203
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