Stratified Interpretation for De Rham Cohomology and Non-Witt Spaces

TL;DR

This work constructs a unified stratified calculus for singular spaces by introducing SCF spaces and a stratified de Rham complex, proving finite-dimensional stratified de Rham cohomology that matches intersection cohomology, and establishing stratified Poincaré duality and a Künneth theorem. It extends the theory to non-Witt spaces through stratified mezzoperversities, linking analytic $L^2$-type complexes with refined intersection homology via quasi-isomorphisms, and obtaining nondegenerate pairings through Verdier duality. Concrete applications to conical singularities and complex curve fibrations illustrate the framework and its compatibility with classical dualities and intersection numbers, while providing a foundation for future exploration of derived-category perspectives and the CGM conjecture. By bridging differential forms, sheaf theory, and intersection homology in a filtered, stratified setting, the paper offers robust tools for analyzing singular geometries and their topological invariants, with potential impact on both theory and applications in birational geometry and beyond.

Abstract

In this paper, we mainly build up the theory of sheaf-correspondence filtered spaces and stratified de Rham complexes for studying singular spaces. We prove the finiteness of a stratified de Rham cohomology and obtain its isomorphism to intersection cohomology through establishing a proper duality theory. Additionally, we present the stratified Poincaré duality, the Künneth decomposition theorem and develop stratified structure theory to non-Witt spaces as an application of a theory of stratified mezzoperversities. Our results connect differential forms, sheaf theory and intersection homology and pave the way for new approaches to study singular geometries, as well as topological invariants on them. Extensions to conical singularities and fibration of complex curves provide examples of the power of this method. This development will be foundational to new tools in stratified calculus and a strengthening of Hodge theory, advancing research in the Cheeger-Goresky-MacPherson conjecture.