Purity of critical cohomology and Kac's conjecture

Ben Davison

Purity of critical cohomology and Kac's conjecture

Representation Theory 2017-03-14 v7 Algebraic Geometry

Authors: Ben Davison

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 a quiver with potential $(\tilde{Q},W)$ associated to $Q$ , which in turn implies positivity of the Kac polynomials for $Q$ .

Keywords

Kac positivity conjecture quiver critical cohomology Donaldson-Thomas theory cohomological integrality theorem dimensional reduction purity Kac polynomials

Cite

@article{arxiv.1311.6989,
  title  = {Purity of critical cohomology and Kac's conjecture},
  author = {Ben Davison},
  journal= {arXiv preprint arXiv:1311.6989},
  year   = {2017}
}

Comments

15 pages. v6: further minor improvements, Lemma 4.4 is cleared up. v7: further minor changes thanks to referee's suggestions

Purity of critical cohomology and Kac's conjecture

Abstract

Keywords

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