Oberseminar: Martin Sera (G.15.25)
Let $X$ be a complex space of pure-dimension $n$. For a pseudoconvex relatively compact domain in $X$ with $C^3$-smooth boundary and embedded in a domain of the complex number space, we prove that the $L^2$- and $L^{2,\mathrm{loc}}$-Dolbeault $(n,q)$-cohomology groups are vanishing for $q>0$. Thereby, we include the case that the forms have values in a Nakano semi-positive holomorphic vector bundle.
Using this local vanishing theorem, we also prove the equivalence of the $L^2$- and $L^{2,\mathrm{loc}}$-Dolbeault $(n,q)$-cohomology groups of relatively compact domains $\Omega=\{\rho<0\}$ in $X$ which are defined by a $C^3$-smooth function $\rho$ which is strictly plurisubharmonic on a neighbourhood of $\partial\Omega$ except of finitely many points.
Those results can be found in the preprint arxiv.org/abs/2605.26700.
