The integral identity conjecture in motivic homotopy theory

TL;DR

This paper extends the functorial approach to the integral identity conjecture in motivic homotopy theory, broadening its scope to more general algebraic spaces with group actions.

Contribution

It generalizes the functorial version of the conjecture to algebraic $S$-spaces with $ au$-locally linearizable $G_m$-actions, building on Ivorra's work.

Findings

Extended the scope of the integral identity conjecture to algebraic $S$-spaces.

Connected the conjecture to $ au$-locally linearizable $G_m$-actions.

Built upon Ivorra's functorial approach using motivic homotopy theory.

Abstract

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 $\tau$-locally linearizable action of $\mathbb{G}_m$ over a noetherian base scheme $S$.