---
title: Middle Perversity Moduli Stack Overview
url: https://www.emergentmind.com/topics/middle-perversity-moduli-stack
type: topic
---

# Middle Perversity Moduli Stack Overview

Searching arXiv for the cited papers to ground the article in current research.
arXiv search query: 2508.17833 OR 2510.23835 OR 2606.27790
The middle perversity moduli stack is a moduli object attached to a stratified space or stack that parametrizes families satisfying middle-perversity conditions in a perverse \(t\)-structure, together with whatever additional structures are built into the ambient theory. In recent arXiv literature, the term occurs in at least three technically distinct but related settings: as the open derived stack of middle-perverse constructible families with perfect stalks on a stratified topological, analytic, or algebraic space [2510.23835, 2606.27790]; as a stratified-microlocal moduli stack of self-dual middle-perversity boundary data on links of singular strata, used to organize \(L^2\)-to-intersection-cohomology comparisons [2508.17833]; and as a small \(v\)-stack parametrizing middle-perverse motivic objects on \(\mathrm{Moduli}_G\) with mixed-parity Hodge enhancements [2311.10019]. The common structure is that middle perversity is enforced fiberwise by stalk/costalk or microlocal conditions, while deformation theory is governed by self-\(\operatorname{Ext}\) groups.

## 1. Terminological scope and principal formulations

The expression “middle perversity moduli stack” does not denote a single universally fixed object across the literature. Rather, it names a family of moduli constructions whose shared feature is the imposition of middle-perversity conditions on constructible or motivic objects relative to a chosen stratification.

| Framework | Ambient setting | Moduli object |
|---|---|---|
| Constructible/perverse moduli | Conically stratified, Whitney stratified, subanalytic, or algebraic spaces | Open derived stack of \(p\)-perverse families, with \(p=p_{\mathrm{mid}}\) [2510.23835, 2606.27790] |
| Stratified-microlocal metric theory | Complex projective varieties with Whitney stratification and admissible stratified Kähler metrics | \(\mathscr{M}^{\mathrm{mid}}=\prod_{Y\in \mathrm{Sing}(X)}\mathrm{LGr}(\mathcal H^{\mathrm{mid}}(L_Y))\) [2508.17833] |
| Mixed-parity \(v\)-stack theory | Small \(v\)-stacks, especially \(\mathrm{Moduli}_G\) over the extended Fargues–Fontaine curve | \(\mathsf{Perv}^{\mathrm{mid}}_{\Lambda,\mathrm{mix}}(\mathrm{Moduli}_G)\) [2311.10019] |

In the constructible-sheaf setting, the central object is an open substack of the Toën–Vaquié moduli of objects. For a stratified space \((X,P)\), one forms the derived stack of constructible complexes with perfect stalks and then cuts out the perverse-flat locus for a chosen perversity \(p\); specializing \(p\) to middle perversity yields the middle perversity moduli stack [2510.23835, 2606.27790]. In the metric-microlocal setting of singular projective varieties, the moduli stack is instead built from link cohomology and Lagrangian boundary conditions, and it organizes the self-dual data needed for a derived enhancement of the Cheeger–Goresky–MacPherson correspondence [2508.17833].

In the \(v\)-stack framework, the same phrase refers to families of objects in the middle-perverse heart of a motivic derived \(\infty\)-category on \(\mathrm{Moduli}_G\), together with mixed-parity \(p\)-adic Hodge structures transported by mixed-parity Riemann–Hilbert functors [2311.10019]. A plausible implication is that the phrase now functions less as a unique definition than as a stable research label for moduli problems centered on middle-perverse hearts.

## 2. Perverse \(t\)-structures and normalization of middle perversity

All formulations depend on a perverse \(t\)-structure determined by a stratification. In the general constructible framework of [2510.23835], if \(i_S:S\hookrightarrow X\) is the inclusion of a stratum and \(p:P\to \mathbb Z\) is a perversity, then an object \(F\in D^b_c(X;R)\) is perverse when
\[
H^j(i_S^*F)=0 \text{ for } j>-p(S), \qquad H^j(i_S^!F)=0 \text{ for } j<-p(S).
\]
The heart \({}^p\mathrm{Perv}_P(X; \mathrm{Mod}_R)\) is abelian, and under the stated finiteness hypotheses the perverse \(t\)-structure restricts to compactly generated constructible sheaves.

The literature represented here uses different normalizations for “middle perversity.” In [2510.23835], on a complex Whitney stratified space \((X,P)\), middle perversity is defined by
\[
p(S)=\dim_{\mathbb C}(S).
\]
In [2606.27790], for the complex algebraic case, the middle perversity is
\[
p_{\mathrm{mid}}(S)=-\dim_{\mathbb C}S,
\]
so that \(\underline{k}_X[n]\) is perverse. In [2508.17833], the metric-intersection-cohomology framework uses the classical lower and upper middle perversities indexed by complex codimension \(c\),
\[
\bar m(c)=\left\lfloor \frac{c-2}{2}\right\rfloor,\qquad \bar n(c)=\left\lceil \frac{c-2}{2}\right\rceil.
\]
These formulas show that the phrase “middle perversity” is convention-dependent; the differences come from sign and shift normalizations rather than from a single contradictory theorem.

The \(\infty\)-categorical hypersheaf treatment of [2606.27790] formulates the perverse \(t\)-structure by
\[
{}^{p}Sh(X;\mathrm{Mod}_A)_{\ge 0}
=\{F\mid i_S^{*}F\in Sh(X_S;\mathrm{Mod}_A)_{\ge p(S)}\},
\]
\[
{}^{p}Sh(X;\mathrm{Mod}_A)_{\le 0}
=\{F\mid i_S^{!}F\in Sh(X_S;\mathrm{Mod}_A)_{\le p(S)}\},
\]
with heart \({}^p\mathrm{Perv}^{\hyp}(X;\mathrm{Mod}_A)\). The passage between these formulations is one of normalization and indexing.

## 3. Derived moduli of middle-perverse objects

A major development is the realization of middle-perverse sheaves as open substacks inside derived moduli of constructible objects. In [2606.27790], for a categorically compact exodromic stratified space with locally weakly contractible strata, the moduli of constructible complexes with perfect stalks is the derived stack
\[
\mathrm{Cons}_P(X)(\operatorname{Spec}A)\simeq
\big(\mathrm{Cons}_{P,\omega}(X;\mathrm{Mod}_A)\big)^{\simeq},
\]
identified with the Toën–Vaquié moduli of objects of \(\mathrm{Cons}_P(X;\mathrm{Mod}_k)\). Exodromy gives
\[
\mathrm{Fun}\big(\Pi_\infty(X,P),\mathrm{Perf}_A\big)\simeq
\mathrm{Cons}_{P,\omega}(X;\mathrm{Mod}_A),
\]
which is the structural mechanism behind representability.

For any perversity \(p\), the locus of \(p\)-perverse families forms a representable open immersion
\[
{}^p\mathbf{Perv}_P(X)\hookrightarrow \mathrm{Cons}_P(X),
\]
and this stack is a derived \(1\)-Artin stack locally of finite presentation. Specializing \(p\) to \(p_{\mathrm{mid}}\) yields the middle perversity moduli stack
\[
{}^{p_{\mathrm{mid}}}\mathbf{Perv}(X)\subset \mathrm{Cons}(X),
\]
in both the compact subanalytic and algebraic cases [2606.27790].

The parallel treatment in [2510.23835] works over a discrete noetherian ring \(k\) of characteristic \(0\) and uses the moduli stack \(\mathcal M_C\) of pseudo-perfect objects in a compactly generated presentable stable \(k\)-linear \(\infty\)-category \(C\). For conically stratified spaces satisfying the stated compactness hypotheses, the perverse flat-locus \({}^p\mathrm{Perv}_P(X)\subset \mathrm{Cons}_P(X)\) is open, and its classical truncation \(t_0{}^p\mathrm{Perv}_P(X)\) is an algebraic \(1\)-Artin stack locally of finite presentation. In that paper, the “Middle Perversity Moduli Stack” is precisely this classical truncation for the middle perversity on a complex Whitney stratified manifold [2510.23835].

At a point represented by a perverse object \(E\) or \(F\), the deformation theory is controlled by self-\(\operatorname{Ext}\) groups. The derived tangent complex is
\[
\mathbb T_E \simeq \mathbf{R}\!\operatorname{Hom}(E,E)[1],
\]
and passage to the classical truncation yields
\[
\operatorname{Aut}(E)\cong \operatorname{Ext}^0(E,E),\qquad
T_E\cong \operatorname{Ext}^1(E,E),\qquad
\text{obstructions in }\operatorname{Ext}^2(E,E)
\]
[2510.23835, 2606.27790].

## 4. Stratified-microlocal and metric formulation

The most specialized use of the term appears in the derived stratified-microlocal framework of “Derived Stratified-Microlocal Framework and Moduli Space Resolution for the Cheeger-Goresky-MacPherson Conjecture” [2508.17833]. There the ambient category is the stratified metric \(\infty\)-category \(\mathbf{StratMet}_\infty\), whose objects are quadruples
\[
(X,\mathcal S,g,\{\Phi_Y\}_{Y\in \mathrm{Sing}(X)}),
\]
where \(X\subseteq \mathbb P^N\) is a complex projective variety or derived scheme, \(\mathcal S\) is a Whitney or derived Whitney stratification, \(g\) is an admissible stratified Kähler metric on \(X_{\mathrm{reg}}\) quasi-isometric to \(ds^2_{\mathrm{FS}}|_{X_{\mathrm{reg}}}\), and each \(\Phi_Y\) is an asymptotic model functor
\[
\Phi_Y:(U_Y,g)\xrightarrow{\sim}
\big(\mathbb R_{>0}\times L_Y\times \mathbb C^m,\;
dr^2+r^{2c}g_{L_Y}+g_{\mathbb C^m}\big).
\]

In this setting, local middle data at a singular stratum \(Y\) is taken from the middle-degree cohomology of the link \(L_Y\). Writing
\[
\mathcal H^{\mathrm{mid}}(L_Y):=H^{d_Y}(L_Y;\mathbb C),
\qquad d_Y=\dim_{\mathbb R}L_Y,
\]
with its Poincaré pairing, the paper defines
\[
\mathscr M_Y:=\mathrm{LGr}\big(\mathcal H^{\mathrm{mid}}(L_Y),\langle-,-\rangle_{L_Y}\big),
\]
and then the middle perversity moduli stack
\[
\mathscr M^{\mathrm{mid}}:=\prod_{Y\in \mathrm{Sing}(X)}\mathscr M_Y.
\]
Its \(S\)-points consist of Lagrangian subbundles \(\mathscr W_{Y,S}\subset \mathcal H^{\mathrm{mid}}(L_Y)\otimes \mathcal O_S\) together with an \(S\)-family of perverse complexes \(E_S\in D^b_c(X\times S)\) satisfying constructibility, fiberwise middle-perversity constraints, and microlocal admissibility.

The microlocal input is a stratified singular characteristic variety
\[
\mathrm{SSH}_{\mathrm{strat}}(E):=
\bigcup_{Y\in \mathrm{Sing}(\mathcal S)}
\big(T^*_Y X\times \mathrm{Cone}(\lambda_Y)\big),
\qquad
\mathrm{Cone}(\lambda_Y)=\{\lambda\in \mathbb R:\lambda\le \lambda_Y\},
\]
which augments Kashiwara–Schapira microsupport by a growth parameter. The admissibility condition
\[
SS(E)\subset \bigcup_Y T_Y^*X,\qquad
\mathrm{SSH}_{\mathrm{strat}}(E)\subset
\bigcup_Y T_Y^*X\times \{\lambda\le \lambda_Y\}
\]
is used to synchronize perverse constraints with \(L^2\)-integrability near strata.

A further central object is the universal truncation complex
\[
\Omega_{X,\mathrm{FS}}^{\bullet,\mathrm{univ}}\in
D^b_c(X\times \mathscr M^{\mathrm{mid}}),
\]
characterized by
\[
\iota_{[E]}^*\Omega_{X,\mathrm{FS}}^{\bullet,\mathrm{univ}}
\simeq {}^{\bar m}\tau_{\le 0}(E).
\]
Fiberwise, its sections are differential forms satisfying middle-perversity growth conditions determined by the tautological Lagrangian subbundles and the Fubini–Study metric. Within this framework the paper states a natural quasi-isomorphism
\[
\Omega_{X,\mathrm{FS}}^{\bullet,\mathrm{univ}}\simeq \mathrm{IC}_X
\]
and hence the isomorphism
\[
H^*_{(2)}\big(X_{\mathrm{reg}},ds^2_{\mathrm{FS}}\big)\cong IH^*(X,\mathbb C),
\]
extending the classical Cheeger–Goresky–MacPherson identification by explicit microlocal growth conditions and without imposing transverse singularity constraints [2508.17833].

## 5. Geometric structure: openness, good moduli, and duality

The geometric behavior of middle perversity moduli stacks differs across frameworks. In the general derived-sheaf setting, openness of the perverse locus is proved by reducing perversity to Tor-amplitude conditions on \(i_S^*F\) and \(i_S^!F\) along strata. This gives a representable open immersion into the constructible moduli stack and yields a derived \(1\)-Artin stack locally of finite presentation [2606.27790]. In the formulation of [2510.23835], the same openness is expressed as the \(\tau\)-flat locus in the Toën–Vaquié moduli of objects, under universal openness of flatness.

The paper [2510.23835] goes further by constructing good moduli spaces in Alper’s sense. For an algebraic stack \(\mathcal X\), a good moduli space is a qcqs morphism \(\phi:\mathcal X\to X\) to an algebraic space such that \(\phi_*\mathcal O_{\mathcal X}\cong \mathcal O_X\) and \(\phi_*\!:\mathrm{QCoh}(\mathcal X)\to \mathrm{QCoh}(X)\) is exact. Using the AHLH criterion, the paper proves that for a complex Whitney stratified manifold with middle perversity, the entire stack \(t_0{}^p\mathrm{Perv}_P(X)\) admits a separated good moduli space. Its \(\kappa\)-points parametrize semisimple perverse sheaves with perfect stalks, and more generally closed points of the stack correspond precisely to semisimple pseudo-perfect objects in the heart [2510.23835].

The metric-microlocal theory of [2508.17833] emphasizes a different geometric feature: parametrized Verdier duality. With
\[
\mathcal D(E):=\mathbf{R}\!\mathcal H om\big(E,\omega_X[\dim X]\big),
\]
the paper defines an involution
\[
\mathcal D:\mathscr M^{\mathrm{mid}}\to \mathscr M^{\mathrm{mid}},\qquad
\mathcal D^2\simeq \mathrm{id}.
\]
Self-duality of the Lagrangian subspaces \(\mathscr W_Y=\mathscr W_Y^\perp\) ensures that both \(\mathrm{IC}_X\) and \(\Omega_{X,\mathrm{FS}}^{\bullet,\mathrm{univ}}\) are preserved by \(\mathcal D\), while the microlocal constraint is compatible with \(SS(\mathcal D E)=SS(E)\).

A recurrent misconception is that “moduli stack” automatically entails a coarse moduli space in the classical algebro-geometric sense. The current literature is more specific. For perverse sheaves, good moduli spaces are proved under the hypotheses listed in [2510.23835], including characteristic \(0\), pseudo-perfectness, and the relevant compactness conditions. For the metric-microlocal construction, the emphasis is instead on derived Artin representability, Ext-quiver charts, and duality, not on an independent coarse moduli theorem [2508.17833].

## 6. Examples, applications, and broader research context

Several standard examples clarify the role of middle perversity. In the smooth case with trivial stratification, middle-perverse sheaves reduce to local systems placed in cohomological degree \(-\dim_{\mathbb C}X\), and the moduli stack becomes the usual character stack:
\[
\mathrm{LocSys}_k(X)\simeq \operatorname{Fun}\big(\Pi_\infty(X),\mathrm{Perf}_k\big),
\]
with rank-\(n\) truncations
\[
\mathrm{LocSys}_{k,n}(X)\simeq
\operatorname{Rep}\big(\pi_1(X),\mathrm{GL}_n\big)\!/\!/\mathrm{GL}_n
\]
[2606.27790]. For punctured Riemann surfaces, the middle perversity stack becomes the perverse character stack, and its Borel–Moore homology carries an associative Hall product constructed by Hecke correspondences [2606.27790].

In the singular-metric setting, [2508.17833] treats normal crossing singularities, isolated cone singularities, and cusp singularities. For simple normal crossings with links given by products of odd-dimensional spheres, the middle-degree cohomology of each link vanishes, so every \(\mathscr M_Y\) is a point and \(\mathscr M^{\mathrm{mid}}\) is trivial. For isolated cone singularities, the moduli stack becomes a product of Lagrangian Grassmannians
\[
\prod_Y \mathrm{LGr}\big(H^{\dim L}(L)\big),
\]
and fibers of the universal truncation complex impose the boundary conditions selecting the corresponding perverse extensions. For cusp singularities, \(\mathscr M_Y\) is a genuine Lagrangian Grassmannian and the growth exponent \(\lambda_Y\) enforces decay of \(L^2\)-harmonic representatives.

The \(v\)-stack formulation of [2311.10019] places middle perversity in a different domain: perverse motivic derived \(\infty\)-categories on \(\mathrm{Moduli}_G\), equipped with the full \(6\)-functor formalism
\[
(f^*,f_*,f_!,f^!,\otimes,\underline{\mathrm{Hom}}).
\]
There the middle perversity is defined by the intrinsic dimension of strata,
\[
p_{\mathrm{middle}}(S)=-\dim S,
\]
and the corresponding middle perversity moduli stack \(\mathsf{Perv}^{\mathrm{mid}}_{\Lambda,\mathrm{mix}}(\mathrm{Moduli}_G)\) parametrizes \(T\)-relative constructible complexes on \(\mathrm{Moduli}_G\times T\) whose fibers lie in the middle-perverse heart and carry mixed-parity Hodge enhancements. The paper connects this to a mixed-parity version of the Fargues–Scholze geometrization of local Langlands and proposes a correspondence with coherent sheaves on the stack of mixed-parity \(L\)-parameters [2311.10019].

The broader research picture is therefore heterogeneous. One strand treats the middle perversity moduli stack as an open derived Artin moduli of perverse-flat constructible families with perfect stalks [2510.23835, 2606.27790]. A second strand uses the same phrase for a product of linkwise Lagrangian Grassmannian stacks controlling self-dual middle-perversity boundary data and \(L^2\)-admissible microlocal growth [2508.17833]. A third transfers the concept to \(v\)-stacks and mixed-parity \(p\)-adic Hodge theory [2311.10019]. What unifies these constructions is not a single formal definition, but the organizing role of middle-perversity conditions in moduli problems for constructible, microlocal, or motivic objects.

Source: https://www.emergentmind.com/topics/middle-perversity-moduli-stack