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

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 $\mathbb{P}^2$. We describe the preimage $\mathrm{gldim}^{-1}(2)$ of the minimal value $2$ of $\mathrm{gldim}$, which 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.