Purity for graded potentials and quantum cluster positivity
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.
Cite
@article{arxiv.1307.3379,
title = {Purity for graded potentials and quantum cluster positivity},
author = {Ben Davison and Davesh Maulik and Joerg Schuermann and Balazs Szendroi},
journal= {arXiv preprint arXiv:1307.3379},
year = {2015}
}
Comments
34pp, many small improvements, to appear in Compositio Math