Integral cohomology of dual boundary complexes is motivic

TL;DR

This paper provides a motivic framework to understand the integral cohomology of dual boundary complexes in algebraic geometry, leading to new insights about their topological properties and applications to the geometric P=W conjecture.

Contribution

It introduces a motivic characterization of the integral cohomology of dual boundary complexes, establishing their contractibility in certain cases and impacting the study of character varieties.

Findings

Dual boundary complex of stably affine space is contractible.

Motivic characterization links cohomology to algebraic motives.

Application to proof of weak geometric P=W conjecture.

Abstract

In this note, we give a motivic characterization of the integral cohomology of dual boundary complexes of smooth quasi-projective complex algebraic varieties. As a corollary, the dual boundary complex of any stably affine space (of positive dimension) is contractible. In a separate paper [Su23], this corollary has been used by the author in his proof of the weak geometric P=W conjecture for very generic $GL_n(\mathbb{C})$-character varieties over any punctured Riemann surfaces.