A Simple Proof of a Theorem by Uhlenbeck and Yau
| dc.creator | Popovici, Dan | |
| dc.date | 2003-11-04 | |
| dc.date.accessioned | 2026-07-07T05:02:27Z | |
| dc.date.available | 2026-07-07T05:02:27Z | |
| dc.description | 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 $L^2_1$ of $L^2$ sections with $L^2$ first order derivatives in the sense of distributions, which satisfies furthermore $(\mathrm{Id}-π)\circ D''π=0$. 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 $\geq 2.$ 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 $\mathrm{Im} π$ 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. | |
| dc.description | 19 pages | |
| dc.identifier | https://arxiv.org/abs/math/0311031 | |
| dc.identifier | http://arxiv.org/abs/math/0311031 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/69058 | |
| dc.subject | Complex Variables | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 14F05 (Primary); 32U40; 31C10(Secondary) | |
| dc.title | A Simple Proof of a Theorem by Uhlenbeck and Yau | |
| dc.type | text |