The Thullen type extension theorem for holomorphic vector bundles with \(L^ 2\)-bounds on curvature
Let \(X\) be a complex manifold and let \(Q\) be a closed nowhere dense subset of \(X\). The phenomenon of extendability of analytic objects from \(X\backslash Q\) to \(X\) is a central topic in modern analytic geometry. In particular, for analytic functions \(f\) there is a variety of extension theorems, all assuming a regularity of the set \(Q\) (e.g. \(Q\) is an analytic set), a dimension condition (e.g. \(\text{codim} (Q)>1)\) and/or a growth control of \(|f|\) near all points of \(Q\). The paper under review is a part of the author's PhD thesis at the Ruhr-University in Bochum. The objects to be extended this time are holomorphic Hermitian vector bundles; control is imposed on the set \(Q\) and on the growth of the curvature of the canonical holomorphic connection. To be more specific, we reproduce the main result: Main Theorem. Let \(X\) be a complex manifold and \(Q\) a closed subset of \(X\) of the form \(Q = \widetilde Q \backslash G\) where either i) \(\widetilde Q\) is an analytic set in \(X\), \(G\) is an open subset of \(X\) having nonempty intersection with every irreducible component of \(Q\) of complex codimension 1; or ii) \(\widetilde Q\) is a connected Lipschitz submanifold of real codimension 2 and \(G\) is an open subset of \(X\) having nonempty intersection with \(\widetilde Q\). Let \(E\) be a holomorphic Hermitian vector bundle over \(X\backslash Q\) with the curvature \(F\). If \(F\in L^2_{\text{loc}}(X)\), then \(E\) extends through \(Q\) as a reflexive sheaf \(\widetilde E\). If, in addition, \(F\in L^p_{\text{loc}} (X)\) with some \(p\geq 3\), then \(\widetilde E\) is a bundle outside \(Q_0\), which is an analytic set in \(X\) with \(\text{codim}_\mathbb{C} Q_0 > p\). The proof is rather technical. Roughly, the main idea is to prove Malgrange's nonlinear Dolbeault-Grothendieck Lemma with bounds. More specifically, these bounds are derived from Hörmander's celebrated \(L^p\)-estimates for the Cauchy-Riemann complex.
scientific article; zbMATH DE number 931970
scientific article