2 AARON BERTRAM, STEFFEN MARCUS, AND JIE WANG There is a natural free action of the additive group C on the stability manifold and nothing interesting happens to the semistable objects under this action (c.f. Section 2). Therefore it is enough, and in fact more natural, to work with the quo- tient manifold and the map p descents to the quotient. The map p is a submersive surjection (but not proper) in the P1 case whereas it is a covering map in the local P1 case. Although we assume familiarity with stability conditions on a derived category throughout this paper, we provide a cursory review in Section 2 for the sake of read- ability. In Section 3 we recast Okada’s description of Stab(D(P1)) through the lens of geometric and non-geometric chambers in the stability manifold. Furthermore, we describe the local homeomorphism p explicitly in this case. Theorem 1.1. The stability manifold Stab(P1)/C is isomorphic to C and the natural map p : Stab(P1)/C P(HomZ(K(P1), C)) P1 is a holomorphic surjective submersion (but not proper). In Section 4 we consider Bridgeland’s main component of the stability manifold Stab(D) of the triangulated subcategory of Db(T P1) consisting of complexes with cohomology supported on the zero section. We provide the expected description of the local homeomorphism p in this setting. Theorem 1.2. The image of p : Stab0(D)/C −→ P(HomZ(K(D), C)) P1 is equal to C \ Z and p is a covering map. The case of local P1 is a restatement of a minor degenerate case of the main theorem in [2, Theorem 1.2]. We do not claim any originality in these results except the expository part and maybe the gluing argument in the proof of Theorem 1.1. We hope our approach provides a more geometric perspective on the structure of the stability manifold of P1 and local P1. 2. Preliminaries For a thorough introduction to stability conditions on triangulated categories, see [3]. We recall the salient definitions of the theory, beginning with that of a Bridgeland stability condition. Familiarity with the theory of triangulated cate- gories, bounded t-structures, and tilting is assumed. 2.1. Stability conditions. Definition 2.1. A numerical pre-stability condition σ on a triangulated cate- gory D consists of a pair (Z, A), where Z = −d + −1 r : K(D)num1 C is a group homomorphism called central charge and A D is the heart of a t- structure, satisfying the following properties: (a) r(E) 0 for all E A (b) if r(E) = 0 and E A nonzero, then d(E) 0. 1 Throughout this paper, we assume the numerical Grothendieck group of D is finite dimensional.
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