Analytic cycles, BottChern forms, and singular sets for the YangMills flow on Kaehler manifolds
Abstract
It is shown that the singular set for the YangMills flow on unstable holomorphic vector bundles over compact Kaehler manifolds is completely determined by the HarderNarasimhanSeshadri filtration of the initial holomorphic bundle. We assign a multiplicity to irreducible top dimensional components of the singular set of a holomorphic bundle with a filtration by saturated subsheaves. We derive a singular BottChern formula relating the second Chern form of a smooth metric on the bundle to the Chern current of an admissible metric on the associated graded sheaf. This is used to show that the multiplicities of the top dimensional bubbling locus defined via the YangMills density agree with the corresponding multiplicities for the HarderNarasimhanSeshadri filtration. The set theoretic equality of singular sets is a consequence.
 Publication:

arXiv eprints
 Pub Date:
 February 2014
 arXiv:
 arXiv:1402.3808
 Bibcode:
 2014arXiv1402.3808S
 Keywords:

 Mathematics  Differential Geometry
 EPrint:
 Lemma 4.2 has been replaced with a more general argument based on Scheja's theorem. Erratum added