Contractibility of space of stability conditions on the projective plane via global dimension function

Abstract

We compute the global dimension function $\mathrm{gldim}$ on the principal component ${\mathrm{Stab}}^{†}({\mathbb{P}}^2)$ of the space of Bridgeland stability conditions on ${\mathbb{P}}^2$. It admits $2$ as the 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 $x\ge 2$ and that ${\mathrm{gldim}}^{-1}(2)$ is contractible.

26 pages, 7 figures. Final version, to appear in Math. Res. Letter