Purity for graded potentials and quantum cluster positivity

Davison, B., Maulik, D., Schürmann, J. and Szendrői, B. (2015) Purity for graded potentials and quantum cluster positivity. Compositio Mathematica, 151(10), pp. 1913-1944. (doi: 10.1112/S0010437X15007332)

Abstract

Consider a smooth quasi-projective variety XX equipped with a C∗C∗-action, and a regular function f:X→Cf:X→C which is C∗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 ff on proper components of the critical locus of ff, generalizing a result of Steenbrink for isolated quasi-homogeneous singularities. Building on work by Kontsevich and 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.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Davison, Nicholas
• Authors: Davison, B., Maulik, D., Schürmann, J., and Szendrői, B.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Compositio Mathematica
• Publisher: Foundation Compositio Mathematica
• ISSN: 0010-437X
• ISSN (Online): 1570-5846
• Published Online: 19 May 2015
• Copyright Holders: Copyright © 2015 The Authors
• First Published: First published in Compositio Mathematica 151(10):1913-1944
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher