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



Yu-Wei Fan (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)

Chunyi Li (Mathematics Institute, University of Warwick, Coventry, United Kingdom)

Wanmin Liu (Department of Mathematics, Uppsala University, Uppsala, Sweden)

Yu Qiu (Yau Mathematical Sciences Center and Department of Mathematical Sciences, Tsinghua University, Beijing, China; and Beijing Institute of Mathematical Sciences and Applications, Yanqi Lake, Beijing, China)


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