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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.

MSC:
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
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[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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