4 AARON BERTRAM, STEFFEN MARCUS, AND JIE WANG mapping a stability condition σ = (Z, A) to its central charge Z. As a consequence, Stab(D) has a manifold structure induced from HomZ(K(D)num, C). Theorem 2.3 does not imply the local homeomorphism p is a covering map. If σ = (Z, P) Stab(D), and Z is another charge near Z, there may not exist σ = (Z , P ) near σ. Neverthelss, we have the following technical result which guarantees the existence of such σ . Theorem 2.4. [3, Theorem 7.1] Let σ = (Z, P) Stab(D), and choose 0 1. If Z HomZ(K(D), C) satisfies (2.2) |Z (E) Z(E)| sin(π)|Z(E)| for every σ-stable E in D, then there is a stability condition τ = (Z , P ) Stab(D) with the distance f(σ, τ) . The stability manifold Stab(D) comes with natural left and right actions by Aut(D) and C respectively. An automorphism Φ Aut(D) induces an automor- phism ϕ of Knum(D). The action of Φ on a stability condition (Z, P) is defined to be (Z ϕ−1, Φ(P)). The action of z C sends (Z, P(·)) to (e−π −1z · Z, P(Re(z)+·)). In particular, tilting a stability condition along μ R corresponds to the action of z = μ and the heart after tilting becomes P(μ, μ + 1]. 3. Stability manifold of P1 In this section we explicitly describe the local homeomorphism p in the case of P1. In this setting, the triangulated category under consideration is D = Db(Coh(P1)). We denote the stability manifold Stab(D) simply by Stab(P1). This space was studied by Okada and computed to be isomorphic to C2 in [12, Theo- rem 1.1]. Okada’s approach was to classify the various hearts and central charges that appear up to the action of Aut(D). We review this classification from a slightly different perspective. 3.1. Constructing stability conditions. We begin by constructing some interesting stability conditions in Stab(P1). In the next subsection, we will see these examples are essentially all the possibilities. Start with the standard example of stability condition with heart A = Coh(P1) and central charge Z(E) = −deg(E) + −1rk(E) as depicted in Figure 1. Since any vector bundle on P1 splits, the stable objects under this stability condition are all the line bundles and the skyscraper sheaf Cx. Figure 1. The standard central charge.
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