The famous Donaldson–Uhlenbeck–Yau Theorem states that any µ–stable holomorphic vector bundle on a compact Kähler manifold admits a Hermitian Yang–Mills (HYM) metric. Bando and Siu extended this result the singular setting, thus establishing to a correspondence between µ–stable reflexive sheaves and singular HYM connections. It is an interesting question to ask: what do Bando and Siu’s singular HYM connections look like near the singular points? Conjecturally, the complex geometry of the underlying sheaf determines gauge-theoretic singularity behavior. I will explain this conjecture in an important special case and discuss in which situations we know this conjecture to hold. This is joint work with H. Sá Earp and A. Jacob.
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