Global complex analysis - around L^2 holomorphic functions

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Let M be a complex manifold, let S be a complex analytic subset of M and let E →M be a holomorphic vector bundle. Concerning the extension problem for holomorphic sections of E from S to M, we obtained the following.Theorem, Let M be a weakly 1-complete Kahler manifold, let (E, h) be a Hermitian holomorphic vector bundle over M, and let (L, b) be a Hermitian holomorphic line bundle over M. For the curvature forms 【encircled H】_h and 【encircled H】_b of h and b, assume that 【encircled H】_h 【greater than or equal】 0 and 【encircled H】_h-εId_E【cross product】【encircled H】_b hold for some ε>0. Then, for any nonzero holomorphic section S of L, the homomorphism s : H^q(M, K_M【cross product】E) →H^q(M, K_M【cross product】E【cross product】L) has a kernel contained in the closure of zero.We obtained also several results on Levi flat hypersurfaces.