Davison, B. (2018) Positivity for quantum cluster algebras. Annals of Mathematics, 187(1), pp. 157-219. (doi: 10.4007/annals.2018.187.1.3)
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Abstract
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 Schiffler. 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.
| Item Type: | Articles |
|---|---|
| Status: | Published |
| Refereed: | Yes |
| Glasgow Author(s) Enlighten ID: | Davison, Nicholas |
| Authors: | Davison, B. |
| College/School: | College of Science and Engineering > School of Mathematics and Statistics |
| Journal Name: | Annals of Mathematics |
| Publisher: | Mathematical Sciences Publishers |
| ISSN: | 0003-486X |
| ISSN (Online): | 1939-8980 |
| Published Online: | 28 December 2017 |
| Copyright Holders: | Copyright © 2018 Department of Mathematics, Princeton University |
| First Published: | First published in Annals of Mathematics 187(1): 157-219 |
| Publisher Policy: | Reproduced in accordance with the publisher copyright policy |
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