arXiv subjects

Khoa Bang Pham

Publications and source records attributed to Khoa Bang Pham.

4 recordsLinked to original sources

arXiv2026

Motivic nearby functors on perverse Nori motives

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.

math.AG↗

arXiv2026

The integral identity conjecture in motivic homotopy theory

The integral identity conjecture of Kontsevich and Soibelman plays an important role in proving the existence of motivic Donaldson-Thomas invariants for three-dimensional noncommutative Calabi-Yau manifolds. There are a number of different formulations of this conjecture in different contexts, and accordingly, there are corresponding solutions to them. The methods devoted to solving this conjecture are diverse, ranging from $\ell$-adic cohomology of rigid analytic varieties to Hrushovski-Kazhdan motivic integration and motivic Fubini theorem for tropicalization maps,... In a recent work, Ivorra deduces a functorial version of the integral identity in the motivic stable homotopy categories of schemes, from the Braden hyperbolic localization theorem. This functorial version concerns Ayoub's nearby cycles functor associated with a $\mathbb{G}_m$-equivariant function $f \colon \mathbb{V}(\mathcal{E}) \longrightarrow \mathbb{A}^1$ on a vector bundle $\mathbb{V}(\mathcal{E})$ over a field of characteristic zero. In the present work, we follow the functorial approach of Ivorra and extend the scope of the original conjecture by Kontsevich and Soibelman by studying more generally the case of $\mathbb{G}_m$-equivariant functions on algebraic $S$-spaces with a $τ$-locally linearizable action of $\mathbb{G}_m$ over a noetherian base scheme $S$.

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arXiv2026

The motivic Satake equivalence using perverse Nori motives

In this article, we develop the theory of stratified perverse Nori motives to prove a refinement of the geometric Satake equivalence of Mirković-Vilonen, for which we call the Nori motivic Satake equivalence, in contrast to the "Tate motivic" Satake equivalence of Richarz-Scholbach.

math.AG↗

arXiv2025

Nearby cycles at infinity as a triangulated functor

For a polynomial function $f \colon \mathbb{C}^n \longrightarrow \mathbb{C}$, it is well-known in singularity theory (after Thom, Pham, Verdier,...) that outside a finite subset of $\mathbb{C}$, the function is a locally trivial $C^{\infty}$-fibration. The minimal such finite set is called the bifurcation set associated with $f$ and determining the bifurcation sets is a difficult task in singularity theory. To such a function, Raibaut attaches a virtual invariant called motivic nearby cycles at infinity. This invariant lives in some Grothendieck ring of varieties and measures the difference between the Euler characteristics of the general fiber and a fixed fiber. In this work, we show that the motivic nearby cycles at infinity admits a functorial version in the context of motivic homotopy theory, called the motivic nearby cycles functors at infinity. The motivic nearby cycles functors at infinity live in the world of motives and hence capture cohomological information (not just Euler characteristics) of singularities at infinity and realizes to Raibaut's construction in the world of virtual motives. Moreover, our construction is universal in the sense that it is applicable to any theory of nearby cycles functors defined in terms of six operations.

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