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

## TL;DR

This paper develops a theory of the $ar{ ext{d}}$-equation on possibly non-reduced analytic spaces, establishing a Dolbeault-Grothendieck lemma and constructing fine sheaves that resolve the structure sheaf using intrinsic Koppelman formulas.

## Contribution

It introduces a new notion of the $ar{ ext{d}}$-equation on non-reduced spaces and constructs a resolution of the structure sheaf via fine sheaves derived from intrinsic Koppelman formulas.

## Key findings

- Established a Dolbeault-Grothendieck lemma for non-reduced spaces
- Constructed fine sheaves of currents resolving the structure sheaf
- Developed intrinsic semi-global Koppelman formulas on $X$

## Abstract

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 $X$.