Étale cohomology of rank one $$\ell $$-adic local systems in positive characteristic

We show that in positive characteristic special loci of deformation spaces of rank one $$\ell $$ ℓ -adic local systems are quasi-linear. From this we deduce the Hard Lefschetz theorem for rank one $$\ell $$ ℓ -adic local systems and a generic vanishing theorem.

Purity for graded potentials and quantum cluster positivity

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.

Examples of compact K-contact manifolds with no Sasakian metric

Using the hard Lefschetz theorem for Sasakian manifolds, we find two examples of compact K-contact nilmanifolds with no compatible Sasakian metric in dimensions 5 and 7, respectively.