Stability conditions on curves

Abstract

Abstract. We study some examples of Bridgeland-Douglas stability conditions on triangulated categories. From one side we give a complete description of the stability manifolds for smooth projective curves of positive genus. From the other side we study stability conditions on triangulated categories generated by an exceptional collection. In the case of the projective line this leads to the connectedness and simply-connectedness of the stability manifold.

Polynomial Bridgeland stability conditions and the large volume limit

“…For a smooth curve C , stability conditions were constructed by Bridgeland [9], and Stab.C / has been described by Macrì [25] and Okada [29]; in Burban and Kreußler [11] the case of singular curves of genus one was considered.…”

Quotient categories, stability conditions, and birational geometry

“…The reader who is only interested in the classification of autoequivalences should read the proofs of Lemma 4.1 and Corollary 4.2 and may skip the rest of this section. The case dim(X) = 1 was already studied by E. Macrì [17] and S. Okada [26]. We will see that our result is a natural generalization of the case of curves of genus g ≥ 1 (see equation (1)).…”

“…The (semi)stable sheaves of phase φ ∈ (0, 1) coincide with the (semi)stable vector bundles of slope − cot(φπ) in the sense of Mumford. E. Macrì [17] has shown that for g ≥ 1 the group GL + (2, R) acts free and transitive on Stab(X), i.e.…”

Lectures on Bridgeland Stability

“…Then Stab(C) ∼ = H × C (= σ 0 · GL + (2, R)) (see [Bri07,Mac07]), where σ 0 is the stability (Coh(C), −d + √ −1 r) in Example 5.12. Let σ = (P, Z) ∈ Stab(C).…”