Frobenius Morphisms and Stability Conditions

Abstract

We generalize Deng–Du’s folding argument, for the bounded derived category \operatorname{\mathcal{D}}(Q) of an acyclic quiver Q , to the finite-dimensional derived category \operatorname{\mathcal{D}}(\Gamma Q) of the Ginzburg algebra \Gamma Q associated to Q . We show that the F -stable category of \operatorname{\mathcal{D}}(\Gamma Q) is equivalent to the finite-dimensional derived category \operatorname{\mathcal{D}}(\Gamma\mathbb{S}) of the Ginzburg algebra \Gamma\mathbb{S} associated to the species \mathbf{\mathbb{S}} , which is folded from Q . If (Q,\mathbf{\mathbb{S}}) is of Dynkin type, we prove that the space \operatorname{Stab}\operatorname{\mathcal{D}}(\mathbf{\mathbb{S}}) of the stability conditions on \operatorname{\mathcal{D}}(\mathbf{\mathbb{S}}) is canonically isomorphic to the space \operatorname{FStab}\operatorname{\mathcal{D}}(Q) of F -stable stability conditions on \operatorname{\mathcal{D}}(Q) . For the case of Ginzburg algebras, we also prove a similar isomorphism between principal components \operatorname{Stab}^{\circ}\operatorname{\mathcal{D}}(\Gamma\mathbb{S}) and \operatorname{FStab}^{\circ}\operatorname{\mathcal{D}}(\Gamma Q) . There are two applications. One is for the space \operatorname{NStab}\operatorname{\mathcal{D}}(\Gamma Q) of numerical stability conditions in \operatorname{Stab}^{\circ}\operatorname{\mathcal{D}}(\Gamma Q) . We show that \operatorname{NStab}\operatorname{\mathcal{D}}(\Gamma Q) consists of \operatorname{Br}Q/\operatorname{Br}\mathbf{\mathbb{S}} many connected components, each of which is isomorphic to \operatorname{Stab}^{\circ}\operatorname{\mathcal{D}}(\Gamma\mathbb{S}) , for (Q,\mathbf{\mathbb{S}}) is of type (A_{3}, B_{2}) or (D_{4}, G_{2}) . The other is that we relate the F -stable stability conditions to Gepner-type stability conditions.