2003/11/04 by Dan Popovici, Popovici, Dan · 1 citation
Mathematics · #14F05 (Primary) #31C10(Secondary) #32U40 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.AG #math.CV #msc:14F05 #msc:32U40
paper · pdf · doi:10.48550/arxiv.math/0311031
19 pages
arxiv created 2003/11/04 · openalex publication_date 2003/11/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A subbundle of a Hermitian vector bundle (E, h) can be metrically and differentiably defined by the orthogonal projection onto this subbundle. A weakly holomorphic subbundle of a Hermitian holomorphic bundle is, by definition, an orthogonal projection π lying in the Sobolev space L21 of L2 sections with L2 first order derivatives in the sense of distributions, which satisfies furthermore (Id-π)∘ D''π=0. We give a new simple proof of the fact that a weakly holomorphic subbundle of (E, h) defines a coherent subsheaf of \cal O(E), that is a holomorphic subbundle of E in the complement of an analytic set of codimension ≥ 2. This result was the crucial technical argument in Uhlenbeck's and Yau's proof of the Kobayashi-Hitchin correspondence on compact Kähler manifolds. We give here a much simpler proof based on current theory. The idea is to construct local meromorphic sections of Im π which locally span the fibers. We first make this construction on every one-dimensional submanifold of X and subsequently extend it via a Hartogs-type theorem of Shiffman's.