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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 $\omega_X^{p}$.