1 results
Purity for graded potentials and quantum cluster positivity
- Part of
-
- Journal:
- Compositio Mathematica / Volume 151 / Issue 10 / October 2015
- Published online by Cambridge University Press:
- 19 May 2015, pp. 1913-1944
- Print publication:
- October 2015
-
- Article
-
- You have access
- Export citation
-
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.
