Mathematical Research Letters

Volume 25 (2018)

Number 2

Purity of critical cohomology and Kac’s conjecture

Pages: 469 – 488

DOI: http://dx.doi.org/10.4310/MRL.2018.v25.n2.a6

Author

Ben Davison (Department of Mathematics and Statistics, Glasgow University, Glasgow Scotland, United Kingdom)

Abstract

We provide a new proof of the Kac positivity conjecture for an arbitrary quiver $Q$. The ingredients are the cohomological integrality theorem in Donaldson–Thomas theory, dimensional reduction, and an easy purity result. These facts imply the purity of the cohomological Donaldson–Thomas invariants for partially nilpotent representations of a quiver with potential $(\widetilde{Q},W)$ associated to $Q$, which in turn implies positivity of the Kac polynomials for $Q$.

Full Text (PDF format)

Received 27 January 2014

Published 5 July 2018