Holomorphic forms, the $\bar{\partial}$-equation, and duality on a reduced complex space

Magazine article, 2015

We study two natural notions of holomorphic forms on a reduced pure $n$-dimensional complex space $X$: sections of the sheaves $\Omega_X^{\bullet}$ of germs of holomorphic forms on $X_{reg}$ that have a holomorphic extension to some ambient complex manifold, and sections of the sheaves $\omega_X^{\bullet}$ introduced by Barlet. We show that $\Omega_X^p$ and $\omega_X^{n-p}$ are Serre dual to each other in a certain sense. We also provide explicit, intrinsic and semi-global Koppelman formulas for the $\bar{\partial}$-equation on $X$ and introduce fine sheaves $\mathscr{A}_X^{p,q}$ and $\mathscr{B}_X^{p,q}$ of $(p,q)$-currents on $X$, that are smooth on $X_{reg}$, such that $(\mathscr{A}_X^{p,\bullet},\bar{\partial})$ is a resolution of $\Om_X^p$ and, if $\Omega_X^{n-p}$ is Cohen-Macaulay, $(\mathscr{B}_X^{p,\bullet},\bar{\partial})$ is a resolution of $\omega_X^{p}$.