Contents Online
Mathematical Research Letters
Volume 30 (2023)
Number 1
Contractibility of space of stability conditions on the projective plane via global dimension function
Pages: 51 – 87
DOI: https://dx.doi.org/10.4310/MRL.2023.v30.n1.a3
Authors
Abstract
We compute the global dimension function $\operatorname{gldim}$ on the principal component $\operatorname{Stab}^\dagger (\mathbb{P}^2)$ of the space of Bridgeland stability conditions on $\mathbb{P}^2$. It admits $2$ as the minimum value and the preimage $\operatorname{gldim}^{-1} (2)$ is contained in the closure $\overline{\operatorname{Stab}^{\operatorname{Geo}} \mathbb{P}^2}$ of the subspace consisting of geometric stability conditions. We show that $\operatorname{gldim}^{-1} [2, x)$ contracts to $\operatorname{gldim}^{-1} (2)$ for any real number $x \geq 2$ and that $\operatorname{gldim}^{-1} (2)$ is contractible.
C. Li is supported by the Royal Society URF\R1\201129 “Stability condition and application in algebraic geometry” and the Leverhulme Trust ECF-2017-222. W. Liu is supported by a grant from the Knut and Alice Wallenberg Foundation. He would like to thank Tobias Ekholm and Ludmil Katzarkov for comments. Y. Qiu is supported by National Key R&D Program of China (No. 2020YFA0713000), Beijing Natural Science Foundation (Grant No. Z180003) and National Natural Science Foundation of China (Grant No. 12031007).
Received 20 November 2020
Received revised 28 August 2022
Accepted 15 September 2022
Published 21 June 2023