Blowing-up Hermitian Yang--Mills connections

Abstract

We investigate hermitian Yang--Mills connections for pullback vector bundles on blow-ups of Kähler manifolds along submanifolds. Under some mild asumptions on the graded object of a simple and semi-stable vector bundle, we provide a necessary and sufficent numerical criterion for the pullback bundle to admit a sequence of hermitian Yang--Mills connections for polarisations that make the exceptional divisor sufficiently small, and show that those connections converge to the pulled back hermitian Yang-Mills connection of the graded object.

21 pages. Previous Lemma 2.5 was wrong and has been fixed. This required a new approach. Sections 4 and 5 are new, and replace former Sections 3-5. New hypothesis in the main statement regarding the graded object and the codimension of the blown-up locus