Purity for graded potentials and quantum cluster positivity

Author: 

Davison, B
Maulik, D
Schuermann, J
Szendroi, B

Publication Date: 

1 January 2015

Journal: 

Compositio Mathematica

Last Updated: 

2020-10-24T23:53:46.127+01:00

Issue: 

10

Volume: 

151

DOI: 

10.1112/S0010437X15007332

page: 

1913-1944

abstract: 

Consider a smooth quasiprojective variety <i>X</i> equipped with a C*-action, and a regular function <i>f</i> : <i>X</i> → 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 <i>f</i> on proper components of the critical locus of <i>f</i>, 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.

Symplectic id: 

517587

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article