Derived Stratifications and Arithmetic Intersection Theory for Varieties with Isolated Singularities

TL;DR

This paper advances the understanding of singular algebraic varieties by developing a stratified de Rham theory using derived geometry, and proves the convergence of harmonic forms, linking $L^2$-cohomology with intersection cohomology.

Contribution

It introduces a unified stratified de Rham framework for singular spaces and resolves a key convergence gap in the Cheeger-Goresky-MacPherson conjecture.

Findings

Harmonic forms converge strongly on varieties with isolated singularities.

$L^2$-cohomology coincides with intersection cohomology.

Unified stratified and derived geometric approach to singular spaces.

Abstract

In this paper, We develop the stratified de Rham theory on singular spaces using modern tools including derived geometry and stratified structures. This work unifies and extends the de Rham theory, Hodge theory, and deformation theory of singular spaces into the frameworks of stratified geometry, $p$-adic geometry, and derived geometry. Additionally, we close a gap in Ohsawa's original proof, concerning the convergence of $L^2$ harmonic forms in the Cheeger-Goresky-MacPherson conjecture for varieties with isolated singularities. Indicating that harmonic forms converge strongly and the $L^2$-cohomology coincides with intersection cohomology.