## USTC Mathematical Physics (Integrable systems) Seminar——Contractibility of space of stability conditions on the projective plane via global dimension function

**Speaker: **Wanmin Liu, Uppsala University

**Time: **Thursday 15:00-16:30 (Beijing Time), June 4, 2020

**Zoom** (725-874-7880, 314159)

**Abstract: **The global dimension function $\mathrm{gldim}$ is a continuous function defined on Bridgeland stability manifold, and it maps a stability condition to a non-negative real number. We compute the global dimension function $\mathrm{gldim}$ on the principal component $\mathrm{Stab}^{\dag}(\mathbb{P}^2)$ of the space of Bridgeland stability conditions on projective plane $\mathbb{P}^2$. It admits $2$ as minimum value and the preimage $\mathrm{gldim}^{-1}(2)$ is contained in the closure $\overline{\mathrm{Stab}^{\mathrm{Geo}}(\mathbb{P}^2)}$ of the subspace consisting of geometric stability conditions. We show that $\mathrm{gldim}^{-1}[2,x)$ contracts to $\mathrm{gldim}^{-1}(2)$ for any real number $x\geq 2$ and that $\mathrm{gldim}^{-1}(2)$ is contractible. This is a joint work with Yu-Wei Fan, Chunyi Li and Yu Qiu. The preprint is available at arXiv:2001.11984.