Semiglobal results for $\overline \partial$ on complex spaces with arbitrary singularities, Part II

Abstract:

We obtain some $L^2$-results for $\overline \partial$ on forms that vanish to high order on the singular set of a complex space. As a consequence of our main theorem we obtain weighted $L^2$-solvability results for compactly supported $\overline \partial$-closed $(p,q)$-forms $(0\le p\le n, 1\le q< n)$ on relatively compact subdomains $\Omega$ of the complex space that satisfy $H^{n-q}(\Omega , \mathcal {S})=0=H^{n-q+1}(\Omega , \mathcal {S})$ for every coherent $\mathcal {O}_X$-module $\mathcal {S}$. The latter result can be used to give an alternate proof of a theorem of Merker and Porten.