Positivity for quantum cluster algebras

• 成果类型: Article
• 署名作者: Davison, Ben
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2018.187.1.3
• 发表日期: 2018
• 页码: 157-219
• 关键词: quiver varieties; REPRESENTATIONS; potentials

摘要

Building on work by Kontsevich, Soibelman, Nagao and Efimov, we prove the positivity of quantum cluster coefficients for all skew-symmetric quantum cluster algebras, via a proof of a conjecture first suggested by Kontsevich on the purity of mixed Hodge structures arising in the theory of cluster mutation of spherical collections in 3-Calabi Yau categories. The result implies positivity, as well as the stronger Lefschetz property conjectured by Efimov, and also the classical positivity conjecture of Fomin and Zelevinsky, recently proved by Lee and Schiffier Closely related to these results is a categorified no exotics type theorem for cohomological Donaldson Thomas invariants, which we discuss and prove in the appendix.