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Consider a smooth quasi-projective variety
equipped with a
-action, and a regular function
-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
on proper components of the critical locus of
, 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.
The Ehrhart polynomials for the class of 0-symmetric convex lattice polytopes in Euclidean n-space ℝn are investigated. It turns out that the roots of the Ehrhart polynomial and Minkowski's successive minima of such polytopes are closely related by their geometric and arithmetic means. It is also shown that the roots of the Ehrhart polynomials of lattice n-polytopes with or without interior lattice points differ essentially. Furthermore, the structure of the roots in the planar case is studied. Here it turns out that their distribution reflects basic properties of lattice polygons.
We prove a generalization to the context of real geometry of an intersection formula for the vanishing cycle functor, which in the complex context is due to Dubson, Lê, Ginsburg and Sabbah (after a conjecture of Deligne). It is also a generalization of similar results of Kashiwara and Schapira, where these authors work with a suitable assumption about the micro-support of the corresponding constructible complex of sheaves. We only use a similar assumption about the support of the corresponding characteristic cycle so that our result can be formulated in the language of constructible functions and Lagrangian cycles.
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