Derived Stratified-Microlocal Framework and Moduli Space Resolution for the Cheeger-Goresky-Macpherson Conjecture

TL;DR

This paper introduces a new stratified microlocal framework and moduli space resolution for the Cheeger-Goresky-Macpherson conjecture, linking metric asymptotics with intersection cohomology in complex algebraic varieties.

Contribution

It develops a stratified microlocal approach and constructs universal complexes, advancing the understanding of singular topology and moduli space dualities beyond previous limitations.

Findings

Established a microlocal correspondence between metric behavior and topology.

Proved an isomorphism between $H_2^*(X_{reg})$ and intersection cohomology.

Developed new paradigms for high-codimension singularity theories.

Abstract

In this paper, We define the stratified metric $\infty$-category $\mathbf{StratMet}_{\infty}$ and the middle perversity moduli stack $\mathscr{M}^{\mathrm{mid}}$. We construct a universal truncation complex $\Omega_{X,\mathrm{FS}}^{\bullet,\mathrm{univ}}$ for a projective variety $X\subseteq\mathbb{P}^N$. By introducing the stratified singular characteristic variety $\mathrm{SSH}_{\mathrm{strat}}$, we establish a microlocal correspondence between metric asymptotic behavior and topology, proving the natural isomorphism $$H_2^*(X_{\mathrm{reg}}, ds_{\mathrm{FS}}^2) \cong IH^*(X,\mathbb{C}).$$ This framework transcends transverse singularity constraints, achieves moduli space parametrized duality, and develops new paradigms for high-codimension singular topology, quantum singularity theory, and $p$-adic Hodge theory.