Journal
COMPOSITIO MATHEMATICA
Volume 151, Issue 10, Pages 1913-1944Publisher
CAMBRIDGE UNIV PRESS
DOI: 10.1112/S0010437X15007332
Keywords
vanishing cycle sheaf; purity of mixed Hodge structure; cluster algebra; quantum cluster positivity
Categories
Funding
- American Institute of Mathematics
- Fondation Sciences Mathematiques de Paris
- DFG [SFB/TR 45, SFB 878]
- NSF [DMS-1159416]
- EPSRC [EP/I033343/1]
- Humboldt Foundation
- Engineering and Physical Sciences Research Council [EP/I033343/1] Funding Source: researchfish
- EPSRC [EP/I033343/1] Funding Source: UKRI
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1159416] Funding Source: National Science Foundation
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Consider a smooth quasi-projective 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 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.
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