Motivic nearby functors on perverse Nori motives

Abstract: In this article, we show that several candidate motivic (unipotent) nearby functors coincide on perverse Nori motives. In particular, the canonical functor defined via the universal abelian factorization admits an expression in terms of the six operations and yields a monodromy sequence on perverse Nori motives. We then deduce the motivic integral identity of Kontsevich-Soibelman for perverse Nori motives. In the appendix, we also prove a version of Beilinson's equivalence for perverse sheaves of geometric origin, which is used repeatedly throughout the article.

• The paper canonically identifies Ayoub’s motivic nearby and unipotent nearby functors with their universal Nori-motivic counterparts, unconditionally and compatibly with Betti realization.
• It constructs a Nori-motivic monodromy sequence with nilpotent monodromy using logarithmic specialization systems, extending nearby-cycle theory beyond the level of abelian categories.
• It proves compatibility with Braden hyperbolic localization and derives the Kontsevich–Soibelman motivic integral identity for suitable \(\mathbb{G}_m\)-equivariant morphisms over perverse Nori motives.

This paper by Khoa Bang Pham establishes that the candidate motivic nearby and unipotent nearby functors on perverse Nori motives — one defined via Ayoub's six-functor formalism on étale motives, the other via the universal abelian factorization property of perverse Nori motives — are canonically isomorphic. As a consequence, the monodromy sequence lifts to Nori motives and the Kontsevich–Soibelman motivic integral identity holds in this setting (2608.17412).

Context and motivation

Let $k \subset \mathbb{C}$ be a subfield and $f\colon X \to \mathbb{A}^1_k$ a morphism of $k$-varieties. For a coefficient system $H$ with a six-functor formalism, Ayoub constructed motivic nearby functors $\Psi_f^A$ and unipotent nearby functors $\Upsilon_f^A$, together with a monodromy operator $N\colon \Upsilon_f^A \to \Upsilon_f^A(-1)$ fitting into a distinguished triangle $$i^*j_* \to \Upsilon_f^A \to \Upsilon_f^A(-1) \to +1$$. Pham had previously shown, in the étale setting, that Braden's hyperbolic localization transformation commutes with these nearby functors whenever they are expressed via six operations; this commutation is the key input to the motivic proof of the Kontsevich–Soibelman integral identity.

The obstruction to extending this to perverse Nori motives is structural. Ivorra–Morel define $MPerv(X)$ as the universal abelian factorization of the Betti realization functor through perverse sheaves:

$$DA(X) \xrightarrow{{}^pH^0_{univ}} MPerv(X) \xrightarrow{rat_X} Perv(X).$$

Tubach has shown that, assuming the standard conjectures, $f\colon X \to \mathbb{A}^1_k$0 is the heart of a perverse motivic $f\colon X \to \mathbb{A}^1_k$1-structure on $f\colon X \to \mathbb{A}^1_k$2, so one expects a full six-functor formalism and a theory of nearby cycles there. The six-functor formalism itself is available from Ivorra–Morel and Terenzi, but nearby functors were not developed in that framework. The naive definition of $f\colon X \to \mathbb{A}^1_k$3 and $f\colon X \to \mathbb{A}^1_k$4 by universal property is possible because $f\colon X \to \mathbb{A}^1_k$5 and $f\colon X \to \mathbb{A}^1_k$6 are perverse $f\colon X \to \mathbb{A}^1_k$7-exact, but it lacks the functorialities needed for the integral identity argument and is defined only at the level of abelian categories.

Main results

The first main theorem states that for any morphism $f\colon X \to \mathbb{A}^1_k$8 there are natural isomorphisms of specialization systems

$f\colon X \to \mathbb{A}^1_k$9

where $k$0 is the logarithmic specialization system built from symmetric powers of the Kummer motive. Consequently there is a Nori-monodromy sequence

$k$1

on $k$2, with $k$3 nilpotent on bounded complexes. The result is unconditional: it does not rely on the standard conjectures. The chain of compatibilities under realizations can be summarized as follows:

Setting	Nearby functors	Monodromy
Étale motives $k$4	$k$5	$k$6
Nori motives $k$7	$k$8	$k$9
Betti sheaves $H$0	$H$1	$H$2

Three nontrivial points are hidden in the theorem. First, since Ayoub's functors are initially defined on ind-objects, the isomorphisms imply they preserve bounded Nori motives. Second, the existence of the Nori-monodromy operator requires showing that the image of the motivic monodromy sequence under Betti realization is the constructible monodromy sequence, and then computing the homotopy fiber of $H$3, which is only defined on abelian categories. Third, the compatibility of monodromy sequences (not merely functors) under Betti realization strengthens Ayoub's earlier compatibility result.

As an immediate corollary, for a $H$4-equivariant $H$5 on a $H$6-variety, the Braden transformation commutes with $H$7, and all arrows in the resulting diagram are isomorphisms on equivariant objects. In particular, for the standard linear $H$8-action on $H$9 with weights $\Psi_f^A$0, one obtains the motivic integral identity

$\Psi_f^A$1

which is precisely the Kontsevich–Soibelman identity in the Nori-motivic setting.

Method: logarithmic specialization systems

The technical core is the introduction of the logarithmic specialization system

$\Psi_f^A$2

where $\Psi_f^A$3 is the Kummer extension $\Psi_f^A$4. This system exists simultaneously on $\Psi_f^A$5, $\Psi_f^A$6, and $\Psi_f^A$7, and is compatible under realization functors, making it the correct intermediary between the motivic and analytic worlds.

Two reduction principles drive the proofs. The semi-stable reduction theorem of Ayoub shows that for smooth $\Psi_f^A$8 with normal-crossings special fiber, the canonical map $\Psi_f^A$9 is an isomorphism. Combined with resolution of singularities and Mayer–Vietoris, this reduces every statement about compact motives to the one-branch case $\Upsilon_f^A$0. The resulting criteria state: if two specialization systems agree on such local models, they agree on all constructible motives; if moreover they commute with direct sums, they agree on all motives. Compactness of $\Upsilon_f^A$1 and nilpotency of the monodromy operator ($\Upsilon_f^A$2, so $\Upsilon_f^A$3 is nilpotent on compact objects and the truncated sequences split for large $\Upsilon_f^A$4) follow from the same reductions.

On complexes of geometric origin, the paper proves $\Upsilon_f^A$5 as specialization systems. The identification with Beilinson's construction uses the fact that $\Upsilon_f^A$6, where $\Upsilon_f^A$7 corresponds to $\Upsilon_f^A$8 under the monodromy correspondence, together with a perversity argument showing $\Upsilon_f^A$9 preserves the perverse heart. The comparison of monodromy operators proceeds by verifying $N\colon \Upsilon_f^A \to \Upsilon_f^A(-1)$0 using the explicit matrix $N\colon \Upsilon_f^A \to \Upsilon_f^A(-1)$1 for the monodromy action on $N\colon \Upsilon_f^A \to \Upsilon_f^A(-1)$2 and the equivariance of the canonical inclusion $N\colon \Upsilon_f^A \to \Upsilon_f^A(-1)$3. This yields compatibility of the full monodromy triangles under Betti realization, strengthening the functor-level compatibility of Ayoub and the partial result of Ivorra–Morel, which did not verify either $N\colon \Upsilon_f^A \to \Upsilon_f^A(-1)$4 or compatibility of sequences.

Nori-theoretic nearby functors

With the analytic and motivic comparisons in place, the universal functors are shown to form specialization systems themselves (base change $N\colon \Upsilon_f^A \to \Upsilon_f^A(-1)$5 for smooth $N\colon \Upsilon_f^A \to \Upsilon_f^A(-1)$6, its adjoint $N\colon \Upsilon_f^A \to \Upsilon_f^A(-1)$7 for proper $N\colon \Upsilon_f^A \to \Upsilon_f^A(-1)$8, duality, external products — the latter relying on Terenzi's perverse $N\colon \Upsilon_f^A \to \Upsilon_f^A(-1)$9-exactness of $$i^*j_* \to \Upsilon_f^A \to \Upsilon_f^A(-1) \to +1$$0). The key lemma is that $$i^*j_* \to \Upsilon_f^A \to \Upsilon_f^A(-1) \to +1$$1 and $$i^*j_* \to \Upsilon_f^A \to \Upsilon_f^A(-1) \to +1$$2 are perverse $$i^*j_* \to \Upsilon_f^A \to \Upsilon_f^A(-1) \to +1$$3-exact on $$i^*j_* \to \Upsilon_f^A \to \Upsilon_f^A(-1) \to +1$$4, proved via conservativity of the Betti realization. Uniqueness of factorizations through the universal abelian category (Terenzi's rigidity results) then forces $$i^*j_* \to \Upsilon_f^A \to \Upsilon_f^A(-1) \to +1$$5 and $$i^*j_* \to \Upsilon_f^A \to \Upsilon_f^A(-1) \to +1$$6.

For the monodromy sequence, the paper uses the Nori-theoretic logarithmic specialization system. A caveat is stated explicitly: the Kummer extension in $$i^*j_* \to \Upsilon_f^A \to \Upsilon_f^A(-1) \to +1$$7 is not claimed to be unique, since the relevant extension group cannot be computed for Nori motives. Nevertheless, the resulting monodromy sequence on $$i^*j_* \to \Upsilon_f^A \to \Upsilon_f^A(-1) \to +1$$8 transfers to $$i^*j_* \to \Upsilon_f^A \to \Upsilon_f^A(-1) \to +1$$9 via the uniqueness of the transformation $MPerv(X)$0 guaranteed by the universal property, giving the desired triangle on $MPerv(X)$1 with nilpotent monodromy on bounded complexes.

Application to the integral identity

The final section proves that Braden transformations commute with $MPerv(X)$2 and $MPerv(X)$3 for $MPerv(X)$4-locally linearizable $MPerv(X)$5-actions. Since the derivator-based description of Ayoub's functors does not obviously extend to diagrams of varieties (the paper notes it is unclear whether $MPerv(X)$6 can be defined for diagrams), a different route is taken: the commutation is established directly for the standard specialization system $MPerv(X)$7 applied to $MPerv(X)$8, combined with a new compatibility of Braden transformations with tensor products, proved via the projection formula and the interchange law. The authors note this tensor-product compatibility was not previously available in the literature. Specializing to the linear attractor/repeller situation yields the integral identity stated above, now valid over perverse Nori motives.

Appendix: Beilinson's equivalence for geometric origin

The appendix proves that the Beilinson realization functor $MPerv(X)$9 is an equivalence, where $$DA(X) \xrightarrow{{}^pH^0_{univ}} MPerv(X) \xrightarrow{rat_X} Perv(X).$$0 is the thick subcategory generated by $$DA(X) \xrightarrow{{}^pH^0_{univ}} MPerv(X) \xrightarrow{rat_X} Perv(X).$$1 for proper $$DA(X) \xrightarrow{{}^pH^0_{univ}} MPerv(X) \xrightarrow{rat_X} Perv(X).$$2. The proof adapts Achar's exposition of Beilinson's argument: stability of $$DA(X) \xrightarrow{{}^pH^0_{univ}} MPerv(X) \xrightarrow{rat_X} Perv(X).$$3 under the six operations, nearby cycles, vanishing cycles, and gluing data is established first; effaceability is reduced to local systems of geometric origin via Artin vanishing and an induction producing affine opens on which $$DA(X) \xrightarrow{{}^pH^0_{univ}} MPerv(X) \xrightarrow{rat_X} Perv(X).$$4 is right effaceable; the induction on composition factors uses that $$DA(X) \xrightarrow{{}^pH^0_{univ}} MPerv(X) \xrightarrow{rat_X} Perv(X).$$5 is Serre, hence noetherian and artinian. This equivalence is used repeatedly throughout the paper, since it licenses working with perverse sheaves of geometric origin as an abelian heart.

Limitations and open questions

Several restrictions are acknowledged. The comparison results are stated for quasi-projective varieties in the functoriality arguments for the universal functors. The uniqueness of the Nori-theoretic Kummer extension remains open, as does the definition of $$DA(X) \xrightarrow{{}^pH^0_{univ}} MPerv(X) \xrightarrow{rat_X} Perv(X).$$6 for diagrams of varieties, which would allow the derivator description of nearby functors in the Nori setting. More broadly, the existence of a perverse motivic $$DA(X) \xrightarrow{{}^pH^0_{univ}} MPerv(X) \xrightarrow{rat_X} Perv(X).$$7-structure on $$DA(X) \xrightarrow{{}^pH^0_{univ}} MPerv(X) \xrightarrow{rat_X} Perv(X).$$8 with heart $$DA(X) \xrightarrow{{}^pH^0_{univ}} MPerv(X) \xrightarrow{rat_X} Perv(X).$$9 is known only conditionally on the standard conjectures (via Tubach); the present results circumvent this assumption but do not resolve it.

Conclusion

The paper completes the program of transporting nearby cycle theory to perverse Nori motives: Ayoub's motivic constructions, the universal constructions from the abelian factorization, and the classical analytic constructions coincide as specialization systems, carry compatible nilpotent monodromy operators, and support the hyperbolic localization formalism. The resulting Kontsevich–Soibelman integral identity over Nori motives places the motivic DT framework on ground independent of the standard conjectures, and the appendix supplies the Beilinson equivalence for perverse sheaves of geometric origin used throughout.