Point Singularities and Bubbling in Degenerations of Rank-Two Bundles on Threefolds

Abstract

We study one-parameter degenerations of rank-two vector bundles on complex threefolds to a rank-two torsion-free sheaf with an isolated point singularity. We prove a rigidity identity: the algebraic bubbling multiplicity of the central fiber equals one half of the Ext-length of the singularity of its reflexive hull. Furthermore, bubbling is forced when a family develops such an isolated point singularity. We use this identity to obtain smoothability obstructions and construct sharp local smoothings. We realize the local example from earlier joint work with Sun as a global degeneration of smooth Hermitian-Yang-Mills connections. Rescaling this degeneration produces a smooth non-flat Hermitian-Yang-Mills connection on $\mathbb C^3$ with density one at infinity, whose tangent cone at infinity has flat connection part and a multiplicity-one line as the blow-up cycle. We also construct smoothings of elementary modifications of projective-cone singularities with explicit algebraic bubbles. These examples give local models for Hermitian-Yang-Mills point bubbling in complex dimension three and distinguish this phenomenon from bubbling along complex codimension-two loci.