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A theorem of $$L^ 2$$ extension of holomorphic sections of a Hermitian bundle. (Un théorème de prolongement $$L^ 2$$ de sections holomorphes d’un fibré hermitien.) (French) Zbl 0789.32015
Let $$Y$$ be a subvariety of a Stein variety $$X$$, defined by a holomorphic section of a vector bundle $$E$$, having generically a differential of maximal rank. We give sufficient conditions on the curvature of a hermitian line bundle $$L$$, for any section of the line bundle $$K_ Y \otimes L \otimes (\text{det} E)^{-1}$$ to extend to a section of $$K_ X \otimes L$$ on $$X$$, with $$L^ 2$$ estimates. When $$X$$ is a projective variety, we get a purely algebraic condition for the restriction morphism $$H^ 0(X,L) \to H^ 0(Y,L)$$ to be surjective.

MSC:
 32Q20 Kähler-Einstein manifolds
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References:
 [1] [A-V] Andreotti, A., Vesentini, E.: Carleman estimates for the Laplace-Beltrami equation on complex manifolds. Publ. Math., Inst. Hautes Étud. Sci.25, 313–362 (1965) · Zbl 0138.06604 [2] [De1] Demailly, J.P.: Scindage holomorphe d’un morphisme de fibrés vectoriels semi-positifs avec estimationsL 2. In: Lelong, P., Skoda, H. (eds.) Seminaire Amiens 1980/81 et Colloque de Wimereux 1981. (Lects. Notes Math., vol. 919, pp. 77–107) Berlin Heidelberg New York: Springer 1982 [3] [De2] Demailly, J.P.: EstimationsL 2 pour l’opérateurd” d’un fibré vectoriel semi-positif audessus d’une variété kählérienne complète. Ann. Sci. Ec. Norm. Supér15, 457–511 (1982) [4] [D-F] Donnelly, H., Fefferman, C.:L 2-cohomology and index theorem for the Bergman metric. Ann. Math.118, 593–618 (1983) · Zbl 0532.58027 [5] [D-X] Donnelly, H., Xavier, F.: On the differential form spectrum of negatively curved Riemann manifolds. Am. Math. J.106, 169–185 (1984) · Zbl 0547.58034 [6] [G-H] Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley 1978 · Zbl 0408.14001 [7] [Hö] Hörmander, L.: An introduction to complex analysis in several variables. Princeton: Van Nostrand 1966 · Zbl 0138.06203 [8] [LP] Le Potier, J.: Annulation de la cohomologie à valeurs dans un fibré vectoriel holomorphe positif de rang quelconque. Math. Ann.218, 35–53 (1975) · Zbl 0313.32037 [9] [O1] Ohsawa, T.: Vanishing theorems on complete Kähler manifolds. Publ. Res. Inst. Math. Sci.,20, 21–38 (1984) · Zbl 0568.32018 [10] [O2] Ohsawa, T.: On the extension of holomorphic functions 2. Publ. Res. Inst. Math. Sci.24, 265–275 (1988) · Zbl 0653.32012 [11] [O-T] Ohsawa, T., Takegoshi, K.: On the extension ofL 2 holomorphic functions. Math. Z.195, 197–204 (1987) · Zbl 0625.32011
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