Purity for graded potentials and quantum cluster positivity

TL;DR

This paper proves the purity of mixed Hodge structures in certain geometric contexts and applies these results to establish the quantum positivity conjecture for cluster mutations across various classes of quivers.

Contribution

It generalizes Steenbrink's result on Hodge structures and proves the quantum positivity conjecture for all quivers with specific properties, extending previous work.

Findings

Proved purity and hard Lefschetz for vanishing cycle cohomology.

Established quantum positivity for quivers with nondegenerate potentials.

Extended positivity results to quivers from surface triangulations.

Abstract

Consider a smooth quasiprojective variety X equipped with a C*-action, and a regular function f: X -> C which is C*-equivariant with respect to a positive weight action on the base. We prove the purity of the mixed Hodge structure and the hard Lefschetz theorem on the cohomology of the vanishing cycle complex of f on proper components of the critical locus of f, generalizing a result of Steenbrink for isolated quasi-homogeneous singularities. Building on work of Kontsevich-Soibelman, Nagao and Efimov, we use this result to prove the quantum positivity conjecture for cluster mutations for all quivers admitting a positively graded nondegenerate potential. We deduce quantum positivity for all quivers of rank at most 4; quivers with nondegenerate potential admitting a cut; and quivers with potential associated to triangulations of surfaces with marked points and nonempty boundary.