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Let $X$ be a complex space of pure dimension. We introduce fine sheaves$\A^X_q$ of $(0,q)$-currents, which coincides with the sheaves of smooth formson the regular part of $X$, so that the associated Dolbeault complex yields aresolution of the structure sheaf $\hol^X$. Our construction is based onintrinsic and quite explicit semi-global Koppelman formulas.

A Dolbeault-Grothendieck lemma on complex spaces via Koppelman formulas