📝 SummarXiv

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.