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Purity for graded potentials and quantum cluster positivity

Part of: Projective and enumerative geometry Arithmetic rings and other special rings Cycles and subschemes Singularities

Published online by Cambridge University Press:  19 May 2015

Ben Davison ,
Davesh Maulik ,
Jörg Schürmann  and
Balázs Szendrői
Ben Davison
Affiliation:
GEOM, EPFL Lausanne, Switzerland email nicholas.davison@epfl.ch
Davesh Maulik
Affiliation:
Department of Mathematics, Columbia University, USA email dmaulik@math.columbia.edu
Jörg Schürmann
Affiliation:
Mathematisches Institut, Universität Münster, Germany email jschuerm@math.uni-muenster.de
Balázs Szendrői
Affiliation:
Mathematical Institute, University of Oxford, UK email szendroi@maths.ox.ac.uk
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Abstract

Consider a smooth quasi-projective variety $X$ equipped with a $\mathbb{C}^{\ast }$-action, and a regular function $f:X\rightarrow \mathbb{C}$ which is $\mathbb{C}^{\ast }$-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.

Keywords

vanishing cycle sheafpurity of mixed Hodge structurecluster algebraquantum cluster positivity

MSC classification

Primary: 14C30: Transcendental methods, Hodge theory , Hodge conjecture 32S35: Mixed Hodge theory of singular varieties 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants 13F60: Cluster algebras
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 10 , October 2015 , pp. 1913 - 1944
Copyright
© The Authors 2015 

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