Derived Stratifications and Arithmetic Intersection Theory for Varieties with Isolated Singularities

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.