A proof of the $\ell$-adic version of the integral identity conjecture for polynomials

TL;DR

This paper proves the $ ext{l}$-adic integral identity conjecture for polynomials, advancing the understanding of motivic Donaldson-Thomas invariants in algebraic geometry.

Contribution

It provides a complete proof of the numerical version of the conjecture for polynomial potentials over algebraically closed fields.

Findings

Verification of the conjecture in the polynomial case.

Application of Berkovich spaces and comparison theorems.

Use of Künneth isomorphism in cohomology.

Abstract

It is well known that the integral identity conjecture is of prime importance in Kontsevich-Soibelman's theory of motivic Donaldson-Thomas invariants for non-commutative Calabi-Yau threfolds. In this article we consider its numerical version and make it a complete demonstration in the case where the potential is a polynomial and the ground field is algebraically closed. The foundamental tool is the Berkovich spaces whose crucial point is how to use the comparison theorem for nearby cycles as well as the K\"{u}nneth isomorphism for cohomology with compact support.