Middle Perversity Moduli Stack Overview

• Middle perversity moduli stack is a derived moduli object that parametrizes perverse families on stratified spaces using fiberwise middle-perversity conditions.
• It unifies diverse frameworks — constructible, microlocal, and motivic — through self-Ext deformation theory and open immersion techniques.
• This structure supports research in perverse sheaf theory, L²-intersection cohomology, and mixed-parity p-adic Hodge enhancements.

Searching arXiv for the cited papers to ground the article in current research. arXiv search query: (Luo, 25 Aug 2025) OR (Lampetti, 27 Oct 2025) OR (Haine et al., 26 Jun 2026) 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 (Lampetti, 27 Oct 2025, Haine et al., 26 Jun 2026); 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 (Luo, 25 Aug 2025); and as a small $v$-stack parametrizing middle-perverse motivic objects on $\mathrm{Moduli}_G$ with mixed-parity Hodge enhancements (Tong, 2023). 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 $p=p_{\mathrm{mid}}$ (Lampetti, 27 Oct 2025, Haine et al., 26 Jun 2026)
Stratified-microlocal metric theory	Complex projective varieties with Whitney stratification and admissible stratified Kähler metrics	$$\mathscr{M}^{\mathrm{mid}}=\prod_{Y\in \mathrm{Sing}(X)}\mathrm{LGr}(\mathcal H^{\mathrm{mid}}(L_Y))$$ (Luo, 25 Aug 2025)
Mixed-parity $v$-stack theory	Small $v$-stacks, especially $L^2$0 over the extended Fargues–Fontaine curve	$L^2$1 (Tong, 2023)

In the constructible-sheaf setting, the central object is an open substack of the Toën–Vaquié moduli of objects. For a stratified space $L^2$2, one forms the derived stack of constructible complexes with perfect stalks and then cuts out the perverse-flat locus for a chosen perversity $L^2$3; specializing $L^2$4 to middle perversity yields the middle perversity moduli stack (Lampetti, 27 Oct 2025, Haine et al., 26 Jun 2026). 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 (Luo, 25 Aug 2025).

In the $L^2$5-stack framework, the same phrase refers to families of objects in the middle-perverse heart of a motivic derived $L^2$6-category on $L^2$7, together with mixed-parity $L^2$8-adic Hodge structures transported by mixed-parity Riemann–Hilbert functors (Tong, 2023). 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 $L^2$9-structures and normalization of middle perversity

All formulations depend on a perverse $v$0-structure determined by a stratification. In the general constructible framework of (Lampetti, 27 Oct 2025), if $v$1 is the inclusion of a stratum and $v$2 is a perversity, then an object $v$3 is perverse when

$v$4

The heart $v$5 is abelian, and under the stated finiteness hypotheses the perverse $v$6-structure restricts to compactly generated constructible sheaves.

The literature represented here uses different normalizations for “middle perversity.” In (Lampetti, 27 Oct 2025), on a complex Whitney stratified space $v$7, middle perversity is defined by

$v$8

In (Haine et al., 26 Jun 2026), for the complex algebraic case, the middle perversity is

$v$9

so that $\mathrm{Moduli}_G$0 is perverse. In (Luo, 25 Aug 2025), the metric-intersection-cohomology framework uses the classical lower and upper middle perversities indexed by complex codimension $\mathrm{Moduli}_G$1,

$\mathrm{Moduli}_G$2

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 $\mathrm{Moduli}_G$3-categorical hypersheaf treatment of (Haine et al., 26 Jun 2026) formulates the perverse $\mathrm{Moduli}_G$4-structure by

$\mathrm{Moduli}_G$5

$\mathrm{Moduli}_G$6

with heart $\mathrm{Moduli}_G$7. 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 (Haine et al., 26 Jun 2026), 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{Moduli}_G$8

identified with the Toën–Vaquié moduli of objects of $\mathrm{Moduli}_G$9. Exodromy gives

$\operatorname{Ext}$0

which is the structural mechanism behind representability.

For any perversity $\operatorname{Ext}$1, the locus of $\operatorname{Ext}$2-perverse families forms a representable open immersion

$\operatorname{Ext}$3

and this stack is a derived $\operatorname{Ext}$4-Artin stack locally of finite presentation. Specializing $\operatorname{Ext}$5 to $\operatorname{Ext}$6 yields the middle perversity moduli stack

$\operatorname{Ext}$7

in both the compact subanalytic and algebraic cases (Haine et al., 26 Jun 2026).

The parallel treatment in (Lampetti, 27 Oct 2025) works over a discrete noetherian ring $\operatorname{Ext}$8 of characteristic $\operatorname{Ext}$9 and uses the moduli stack $p$0 of pseudo-perfect objects in a compactly generated presentable stable $p$1-linear $p$2-category $p$3. For conically stratified spaces satisfying the stated compactness hypotheses, the perverse flat-locus $p$4 is open, and its classical truncation $p$5 is an algebraic $p$6-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 (Lampetti, 27 Oct 2025).

At a point represented by a perverse object $p$7 or $p$8, the deformation theory is controlled by self-$p$9 groups. The derived tangent complex is

$p=p_{\mathrm{mid}}$0

and passage to the classical truncation yields

$p=p_{\mathrm{mid}}$1

(Lampetti, 27 Oct 2025, Haine et al., 26 Jun 2026).

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” (Luo, 25 Aug 2025). There the ambient category is the stratified metric $p=p_{\mathrm{mid}}$2-category $p=p_{\mathrm{mid}}$3, whose objects are quadruples

$p=p_{\mathrm{mid}}$4

where $p=p_{\mathrm{mid}}$5 is a complex projective variety or derived scheme, $p=p_{\mathrm{mid}}$6 is a Whitney or derived Whitney stratification, $p=p_{\mathrm{mid}}$7 is an admissible stratified Kähler metric on $p=p_{\mathrm{mid}}$8 quasi-isometric to $p=p_{\mathrm{mid}}$9, and each $$\mathscr{M}^{\mathrm{mid}}=\prod_{Y\in \mathrm{Sing}(X)}\mathrm{LGr}(\mathcal H^{\mathrm{mid}}(L_Y))$$0 is an asymptotic model functor

$$\mathscr{M}^{\mathrm{mid}}=\prod_{Y\in \mathrm{Sing}(X)}\mathrm{LGr}(\mathcal H^{\mathrm{mid}}(L_Y))$$1

In this setting, local middle data at a singular stratum $$\mathscr{M}^{\mathrm{mid}}=\prod_{Y\in \mathrm{Sing}(X)}\mathrm{LGr}(\mathcal H^{\mathrm{mid}}(L_Y))$$2 is taken from the middle-degree cohomology of the link $$\mathscr{M}^{\mathrm{mid}}=\prod_{Y\in \mathrm{Sing}(X)}\mathrm{LGr}(\mathcal H^{\mathrm{mid}}(L_Y))$$3. Writing

$$\mathscr{M}^{\mathrm{mid}}=\prod_{Y\in \mathrm{Sing}(X)}\mathrm{LGr}(\mathcal H^{\mathrm{mid}}(L_Y))$$4

with its Poincaré pairing, the paper defines

$$\mathscr{M}^{\mathrm{mid}}=\prod_{Y\in \mathrm{Sing}(X)}\mathrm{LGr}(\mathcal H^{\mathrm{mid}}(L_Y))$$5

and then the middle perversity moduli stack

$$\mathscr{M}^{\mathrm{mid}}=\prod_{Y\in \mathrm{Sing}(X)}\mathrm{LGr}(\mathcal H^{\mathrm{mid}}(L_Y))$$6

Its $$\mathscr{M}^{\mathrm{mid}}=\prod_{Y\in \mathrm{Sing}(X)}\mathrm{LGr}(\mathcal H^{\mathrm{mid}}(L_Y))$$7-points consist of Lagrangian subbundles $$\mathscr{M}^{\mathrm{mid}}=\prod_{Y\in \mathrm{Sing}(X)}\mathrm{LGr}(\mathcal H^{\mathrm{mid}}(L_Y))$$8 together with an $$\mathscr{M}^{\mathrm{mid}}=\prod_{Y\in \mathrm{Sing}(X)}\mathrm{LGr}(\mathcal H^{\mathrm{mid}}(L_Y))$$9-family of perverse complexes $v$0 satisfying constructibility, fiberwise middle-perversity constraints, and microlocal admissibility.

The microlocal input is a stratified singular characteristic variety

$v$1

which augments Kashiwara–Schapira microsupport by a growth parameter. The admissibility condition

$v$2

is used to synchronize perverse constraints with $v$3-integrability near strata.

A further central object is the universal truncation complex

$v$4

characterized by

$v$5

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

$v$6

and hence the isomorphism

$v$7

extending the classical Cheeger–Goresky–MacPherson identification by explicit microlocal growth conditions and without imposing transverse singularity constraints (Luo, 25 Aug 2025).

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 $v$8 and $v$9 along strata. This gives a representable open immersion into the constructible moduli stack and yields a derived $v$0-Artin stack locally of finite presentation (Haine et al., 26 Jun 2026). In the formulation of (Lampetti, 27 Oct 2025), the same openness is expressed as the $v$1-flat locus in the Toën–Vaquié moduli of objects, under universal openness of flatness.

The paper (Lampetti, 27 Oct 2025) goes further by constructing good moduli spaces in Alper’s sense. For an algebraic stack $v$2, a good moduli space is a qcqs morphism $v$3 to an algebraic space such that $v$4 and $v$5 is exact. Using the AHLH criterion, the paper proves that for a complex Whitney stratified manifold with middle perversity, the entire stack $v$6 admits a separated good moduli space. Its $v$7-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 (Lampetti, 27 Oct 2025).

The metric-microlocal theory of (Luo, 25 Aug 2025) emphasizes a different geometric feature: parametrized Verdier duality. With

$v$8

the paper defines an involution

$v$9

Self-duality of the Lagrangian subspaces $L^2$00 ensures that both $L^2$01 and $L^2$02 are preserved by $L^2$03, while the microlocal constraint is compatible with $L^2$04.

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 (Lampetti, 27 Oct 2025), including characteristic $L^2$05, 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 (Luo, 25 Aug 2025).

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 $L^2$06, and the moduli stack becomes the usual character stack: $L^2$07 with rank-$L^2$08 truncations

$L^2$09

(Haine et al., 26 Jun 2026). 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 (Haine et al., 26 Jun 2026).

In the singular-metric setting, (Luo, 25 Aug 2025) 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 $L^2$10 is a point and $L^2$11 is trivial. For isolated cone singularities, the moduli stack becomes a product of Lagrangian Grassmannians

$L^2$12

and fibers of the universal truncation complex impose the boundary conditions selecting the corresponding perverse extensions. For cusp singularities, $L^2$13 is a genuine Lagrangian Grassmannian and the growth exponent $L^2$14 enforces decay of $L^2$15-harmonic representatives.

The $L^2$16-stack formulation of (Tong, 2023) places middle perversity in a different domain: perverse motivic derived $L^2$17-categories on $L^2$18, equipped with the full $L^2$19-functor formalism

$L^2$20

There the middle perversity is defined by the intrinsic dimension of strata,

$L^2$21

and the corresponding middle perversity moduli stack $L^2$22 parametrizes $L^2$23-relative constructible complexes on $L^2$24 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^2$25-parameters (Tong, 2023).

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 (Lampetti, 27 Oct 2025, Haine et al., 26 Jun 2026). 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$26-admissible microlocal growth (Luo, 25 Aug 2025). A third transfers the concept to $L^2$27-stacks and mixed-parity $L^2$28-adic Hodge theory (Tong, 2023). 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.