The Global Dimension Function on Stability Manifolds

Abstract: In this paper, we study the reachability and boundary behavior of the global dimension function on Bridgeland stability manifolds. For a smooth projective variety $X$ of dimension $n$, we prove that the infimum of the global dimension function is $n$. This value is attained when $-K_X$ is ample, is taken by every stability condition when $K_X\in\mathrm{Pic}<sup>0(X)$, and is not attained when $K_X$ is big and nef. We also extend these reachability results to pre-stability conditions. In addition, we construct compactifications of reduced stability spaces of smooth projective curves to which the global dimension function extends continuously. Finally, we discuss algebraic models of the reachability problem, including finite-dimensional algebras and semiorthogonal decompositions.