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Let X be an analytic subvariety of complex Euclidean space with isolated singularity at the origin, and let η be a smooth form of type (1.1) defined on X \ {0}. The main result of this note is a criterion for solubility of the equation ∂∂u = η. This implies a criterion for triviality of a Hermitian-holomorphic line bundle (L, h) → X \ {0} in a neighbourhood of the origin.

A {̄∂}∂–Poincaré lemma for forms near an isolated complex singularity