Point Singularities and Local Third Chern Classes for Rank-Two Torsion-free Sheaves on Threefolds

In this paper, motivated by singularity formation in gauge theory, we study the local third Chern class contribution carried by isolated point singularities of rank-two torsion-free sheaves on complex threefolds. In the local rank-two setting considered here, the invariant is defined in terms of finite-length local algebraic data at the singular point. We prove that it can be computed from data on the total family; in particular, it is deformation invariant. We also prove that its parity recovers a topological invariant of the underlying smooth complex rank-two vector bundle on the boundary sphere. We then give a relative K-theoretic interpretation: a self-dual complex naturally associated with the sheaf defines a local $K$-theoretic charge, and this charge is equal to the local third Chern class. For rank-two reflexive sheaves, we relate the same invariant to several classical algebraic quantities, including the Fitting scheme and the Buchsbaum-Rim multiplicity. We also discuss applications to the boundary of moduli spaces of Hermitian-Yang-Mills connections.