A theorem of \(L^ 2\) extension of holomorphic sections of a Hermitian bundle (Q1319298)

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.