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 $\mathbb{ P}^2$. It admits $2$ as the minimum value and the preimage $\mathrm{ gldim}^{ -1}(2)$ is contained in the closure $\bar{ \mathrm{ Stab}^{ \mathrm{ Geo}}(\mathbb{ P}^2)}$ 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.
Categories: math.AG math.GN math.RT