The Space of Stability Conditions on the Projective Plane

The space of Bridgeland stability conditions on the bounded derived category of coherent sheaves on P2 has a principle connected component Stab(P2). We show that Stab†(P2) is the union of geometric and algebraic stability conditions. As a consequence, we give a cell decomposition for Stab (P2) and show that Stab†(P2) is contractible. Introduction Motivated by the concept of Π-stability condition on string theory by Douglas, the notion of a stability condition, σ = (P, Z), on a C-linear triangulated category T was first introduced by Bridgeland in [Br07]. In the notion, the central charge Z is a group homomorphism from the numerical Grothendieck group K0(T ) to C. Bridgeland proves that the space of stability conditions inherits a natural complex manifold structure via local charts of central charges in HomZ(K0(T ),C). In particular, when K0(T ) has finite rank, the space of stability condition (satisfying support condition), Stab(T ), has complex dimension rank(K0(T )). As mentioned in [Br09], Stab(T ) is expected to be related to the study of string theory and mirror symmetry. The main interesting example is to understand the space of stability conditions on a compact Calabi-Yau threefold X such as a quintic in P4. Yet this problem is still wildly open mainly due to some technical difficulties. Although the compact Calabi-Yau threefold case is still difficult to study, Stab(T ) of various analog categories has been very well understood, see [BSW15, BQS14, DK16, Ik14, Qi15]. While most of these examples are build from quivers or locally derived category of sub-varieties, few cases of Stab(X) for smooth compact varieties X are known. Such Stab(X) is ‘well-understood’ only when X is P1([Ok06]), a curve ([Br07]), a K3 surface ([Br08, BB13]), an abelian surface or threefold ([BMS]). In this paper, based on some important technical results from [Ma04] and [Ma07], we make an attempt to analyze the space Stab(P2). Theorem 0.1 (Theorem 3.9, Corollary 3.10). Let Stab(P2) be the connected component in Stab(P2) that contains the geometric stability conditions, then Stab(P) = Stab(P) ⋃ Stab(P). In particular, Stab(P2) is contractible.