# The $$\bar{\partial }$$ -equation on a non-reduced analytic space

Mathematische Annalen, Apr 2018

Let X be a, possibly non-reduced, analytic space of pure dimension. We introduce a notion of $$\overline{\partial }$$-equation on X and prove a Dolbeault–Grothendieck lemma. We obtain fine sheaves $$\mathcal {A}_X^q$$ of (0, q)-currents, so that the associated Dolbeault complex yields a resolution of the structure sheaf $$\mathscr {O}_X$$. Our construction is based on intrinsic semi-global Koppelman formulas on X.

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Mats Andersson, Richard Lärkäng. The $$\bar{\partial }$$ -equation on a non-reduced analytic space, Mathematische Annalen, 2018, 1-47, DOI: 10.1007/s00208-018-1678-8