Stratified Perverse Nori Motives

• Stratified perverse Nori motives are a universal, abelian, and Q-linear enhancement of perverse sheaves, constructed from the stable homotopy category of constructible étale motives and Betti realization.
• They employ gluing along finite algebraic stratifications and support a closed symmetric monoidal structure with a comprehensive six-functor formalism to ensure compatibility with classical cohomological tools.
• The framework incorporates a canonical weight filtration leading to purity and semisimplicity, with applications including motivic Satake equivalence, motivic Chern classes, and refined Hard Lefschetz theorems.

Stratified perverse Nori motives constitute a universal, abelian, and Q-linear enhancement of the theory of perverse sheaves, built from the stable homotopy category of constructible étale motives via the Nori-Ivorra–Morel–Terenzi formalism. These categories are canonically equipped with gluing structures along finite algebraic stratifications, admit closed symmetric monoidal structures, and carry six operations compatible with classical functoriality and cohomological tools. The stratified setup crucially underpins motivic analogues of the geometric Satake equivalence, weight theory, Tannakian formalism for motivic local systems, motivic Chern classes, and motivic versions of intersection cohomology and Hard Lefschetz theorems.

1. Construction and Gluing of Stratified Perverse Nori Motives

Given a field $k$ of characteristic $0$ with $\sigma:k\hookrightarrow\mathbb{C}$ and a quasi-projective $k$-variety $X$ endowed with a finite Whitney–Nori (or algebraic) stratification

$X = \bigsqcup_{w \in W} X_w,$

where each $X_w$ is smooth and quasi-projective and strata closures are unions of strata, one begins with the triangulated category of constructible étale motives, $DA(X,\Q)$, as defined by Cisinski–Déglise or Ayoub. The Betti realization functor

$Betti^*: DA(X, \Q) \to D^b_{ct}(X^{an}, \Q)$

is composed with perverse cohomology in degree zero to obtain an exact additive functor to perverse sheaves, enabling the definition of the perverse universal abelian hull via the Freyd–Nori construction:

$MPerv(X) := \mathcal{A} \left( DA(X, \Q) \to Perv(X^{an}, \Q) \right),$

where $MPerv(X)$ is abelian, equipped with a faithful exact realization $rat_X : MPerv(X) \to Perv(X^{an}, \Q)$ and is the heart of a motivic perverse t-structure on $DN^b(X) := D^b(MPerv(X))$ (Ivorra et al., 2019, Terenzi, 2024).

The stratified category is then defined by imposing local system conditions stratum-wise:

$$DN^b(X, X^+) := \{ M \in DN^b(X) \mid {}^n H^*(\iota_w^* M) \in MLoc(X_w)[-\dim X_w], \ \forall w,n \}$$

with perverse and constructible t-structure support conditions:

• $^n(\iota_w^*M) = 0$ for $n > -\dim X_w$
• $^n(\iota_w^!M) = 0$ for $n < -\dim X_w$

By gluing/recollement (Beilinson’s formalism), these truncation conditions define a t-structure whose heart,

$$MPerv(X, X^+) := DN^b(X, X^+)^{\leq 0} \cap DN^b(X, X^+)^{\geq 0},$$

is the abelian category of stratified perverse Nori motives. This subcategory is Noetherian, Artinian, and every object carries a canonical weight filtration (Pham, 5 Jan 2026).

2. Functoriality: Six-Operations Formalism and Gluing

The assignment $X \mapsto D^b(MPerv(X))$ admits the full six-functor formalism:

• $f^*, f_*, f_!, f^!, \otimes, \mathrm{Hom}$ where $f$ denotes a morphism of quasi-projective varieties and the external tensor product is

$$\boxtimes: D^b(MPerv(X_1)) \times D^b(MPerv(X_2)) \to D^b(MPerv(X_1 \times X_2))$$

To pass to the stratified category, functors are checked stratum-wise for t-exactness, reducing to affine and closed immersion cases; compatibility with Betti realization ensures that $f^*[d]$ and $f_![−d]$ are t-exact for $f$ smooth of relative dimension $d$ (Ivorra et al., 2019, Terenzi, 2024).

Intermediate extension and standard recollement are established by

$$j_{!*}(M) = \mathrm{Im}\left( {}^pH^0(j_! M) \to {}^pH^0(j_* M) \right)$$

where $j: U \hookrightarrow X$ is a stratum inclusion, mirroring the classical formalism. Adjunctions and base-change/exchange isomorphisms are available, including motivic versions of nearby and vanishing cycles via Ayoub’s specialization system (Ivorra et al., 2019).

3. Weight Filtration, Purity, and Semisimplicity

Every $M \in MPerv(X)$ is endowed with an increasing finite weight filtration $W_\bullet M$, compatible with the motivic, Betti, and $\ell$-adic realizations. The graded pieces $\mathrm{Gr}^W_w M$ are pure of weight $w$, and the pure subcategories $MPerv(X)_w$ are semisimple abelian (Pham, 5 Jan 2026, Ivorra et al., 2019).

Compatibility with $\ell$-adic perverse sheaves via the realization functor ensures that notions of weights and purity are independent of the realization theory. The Chow weight structure on $DA(X,\Q)$ induces the motivic weight filtration, and semisimplicity of pure objects parallels Deligne’s theory. Applications include the arithmetic proof of the purity of intersection cohomology:

$$IH^i(X;\mathcal{L}) = H^{i-\dim X}( S_*^M ( j_{!*}^M \mathcal{L}[\dim X] ) )$$

which carries pure motives of weight $i + w$ (Ivorra et al., 2019).

4. Tannakian Local Systems and Monoidal Structure

On each smooth stratum $X_w$, objects whose Betti realization are shifted local systems form a neutral Tannakian category $MLoc(X_w) \subset MPerv(X)$, with unit $\Q_{X_w}[\dim X_w]$ and tensor structure induced via shifted tensors. These Tannakian categories glue along the stratification to produce global subcategories governed by the motivic Galois group. The closed symmetric monoidal structure on $D^b(MPerv(X))$ is constructed via external and internal tensor products and internal Hom using the universal property and Ind-extension, leading to a well-behaved Tannakian formalism for motivic local systems (Terenzi, 2024).

Convolution constructions, stratified categoricity, and parity vanishing arguments ensure semisimplicity and monoidal compatibility in applications to representation theory.

5. The Nori Motivic Satake Equivalence and Applications

For $G$ a split connected reductive group over $k$, the affine Grassmannian $Gr_G$ admits a stratification by $L^+G$-orbits, yielding the equivariant perverse Nori motive category $MPerv_{L^+G}(Gr_G)$ of pure weight zero objects. The main theorem establishes an equivalence

$MPerv_{L^+G}(Gr_G) \simeq \mathrm{Rep}_\Q^{fd}(G^\vee_\Q \times G_{mot}(k))$

where $G^\vee_\Q$ is the Langlands dual group and $G_{mot}(k)$ is the Nori motivic Galois group. The tensor product is defined by convolution,

$M \star N := m_*(M \widetilde{\boxtimes} N),$

and the category is neutral Tannakian under the motivic fiber functor. Stratification is essential in classifying simples as $IC_\lambda(L)$ for motivic local systems $L$ on smooth strata $Gr^\lambda_G$; convolution preserves motivic purity and stratified structure (Pham, 5 Jan 2026).

This equivalence refines earlier constructions (Richarz–Scholbach’s Tate-motivic Satake equivalence) and generalizes to categories of motivic intersection complexes, equivariant settings, and induces new approaches to motivic Galois group Hopf algebras. Specializations recover Artin and pure Satake categories, with dual groups accordingly.

6. Motivic Chern Classes and Hard Lefschetz

Motivic Chern classes are defined via

$c_i : K_0(MPerv(X)) \to H^{2i}_{Betti}(X, \Q(i))$

by lifting classes of motives to Betti cohomology, compatible with restriction to strata:

$c_i([\mathcal{M}])|_{X_w} = c_i([\mathcal{M}|_{X_w}]) \in H^{2i}(X_w, \Q(i)).$

The relative Hard Lefschetz theorem holds for $f: X \to Y$ projective, stratified compatibly, and $M \in MPerv(X)$,

$$L^i: {}^p R^{-i}f_*(M)\xrightarrow{\cup\,c_1(\mathcal{L})^i}{} {}^p R^i f_*(M)(i)$$

where $(i)$ is Tate twist, verified stratum by stratum via the weight filtration and comparison with perverse sheaves (Terenzi, 2024).

7. Explicit Examples and Compatibility

A classical example is $\mathbb{A}^1_k$ stratified by $U = \mathbb{G}_m$ and $Z = \{0\}$, the intermediate extension

$IC_X(\Q_U) = j_{!*}^M(\Q_U[1]) \in MPerv(\mathbb{A}^1_k)$

fitting into canonical short exact sequences reflecting intersection cohomology’s motivic purity; for nodal plane cubic $C$ with normalization $\nu: \mathbb{P}^1 \to C$, $\nu_*^M \Q_{\mathbb{P}^1}[1]$ splits as $IC_C(\Q) \oplus$ skyscraper at the node, and $IH^*(C)$ is pure of the correct weights (Ivorra et al., 2019).

All constructions are independent of the complex embedding, and up to equivalence, the category over $\mathrm{Spec}(k)$ coincides with Nori’s mixed motives, establishing canonical comparison diagrams and dualities between motivic and classical Galois groups (Pham, 5 Jan 2026, Terenzi, 2024, Ivorra et al., 2019).