Contemporary Mathematics Volume 647, 2015 http://dx.doi.org/10.1090/conm/647/12956 The stability manifolds of P1 and local P1 Aaron Bertram, Steffen Marcus, and Jie Wang Dedicated to Herb Clemens Abstract. In this expository paper, we review and compare the local home- omorphisms from the manifold of Bridgeland stability conditions to the space of central charges in the cases of both P1 and local P1. 1. Introduction The space of stability conditions on a triangulated category was introduced by Bridgeland in [3], following work on Π-stability in string theory due to Douglas [7]. A central result of [3] is that for an arbitrary triangulated category D the space Stab(D) of stability conditions is a topological manifold. Under nice conditions on D, the manifold Stab(D) is in fact a finite dimensional complex manifold. When working with an algebraic variety X, Bridgeland’s stability conditions allow one to abstract notions of slope stability for sheaves on projective varieties to the bounded derived category of coherent sheaves on X. Since [3], there have been many efforts towards computing explicit examples in order to better understand the theory and its relation to enumerative geometry and mirror symmetry. For a smooth curve C of genus at least one, there is a rather uniform and complete description of Stab(D(C)). Macr` ı [11] and Bridgeland [3] proved that they are all isomorphic to the universal cover GL + (2, R) of GL+(2, R). The case of C = P1 turns out to be more subtle and involved due to the existence of full exceptional collections in D(P1). There exists “pathological” regions in Stab(D(P1)) where line bundles are unstable. Okada [12] proved that Stab(D(P1)) C2. In [2], Bridgeland describes a connected component of Stab(D(X)) when X is the minimal resolution of a Kleinian quotient singularity C2/G. In particular, for G = Z/2Z this includes the local case of P1 embedded as the zero section in the total space of its cotangent bundle. The aim of this expository note is to provide a thorough review and comparison of the two cases of P1 and local P1, with emphasis on the local homeomorphisms p from the space of stability conditions to the space of central charges. Through these explicit examples, the authors hope to illustrate some general methods to study stability manifolds. Some of these techniques can be applied to study more complicated examples. Bayer and Macr` ı [6] gave a fairly explicit description of the stability manifold of local P2. c 2015 American Mathematical Society 1
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