A Dolbeault-Grothendieck Resolution for Singular Spaces

TL;DR

This paper generalizes the Dolbeault-Grothendieck resolution to singular complex spaces, providing a functorial construction with applications to cohomology and flat resolutions.

Contribution

It introduces a new resolution for singular spaces that extends classical methods and maintains functoriality and topological properties similar to the smooth case.

Findings

Constructed a Dolbeault-Grothendieck resolution for singular spaces.

Established functorial pullback mappings between resolutions.

Provided topological structures on global sections and cohomology groups.

Abstract

We construct a generalization of the Dolbeault-Grothendieck resolution on a singular complex space. The same construction yields, for each morphism of analytic spaces, a pullback mapping between the respective Dolbeault-Grothendieck resolutions. As in the smooth case, the terms of the resolution are soft sheaves with stalks which are flat with respect to the sheaf of holomorphic sections. If, moreover, the complex space $(X,\mathcal{O}_{X})$ is countable at infinity then the global section spaces of the terms of the resolution are endowed with natural Fr\'{e}chet-Schwarz topologies which induce the natural topology on the cohomology groups $H^{\bullet }(X,\mathcal{O}_{X})$. The construction is an exercise in globalization using semi-simplicial techniques. Using the above construction one can produce, for instance, a soft resolution with $\mathcal{O}_{X}$-flat stalks for the de Rham complex on the analytic space $X$.