Bridgeland stability condition

Motivation To construct well-behaved moduli spaces of vector bundles on a smooth algebraic curve, it is necessary to restrict attention to the class of semistable vector bundles -- these are bundles W that satisfy the inequality \mu(W)\geq \mu(V) for all sub-bundles V\subset W, where \mu(-)=\operatorname{deg}(-)/\operatorname{rank}(-)is the slope function. This inequality of slopes can be equivalently stated by saying the complex number Z(W)=-\operatorname{deg}(W)+i\operatorname{rank}(W) has a greater argument than analogously defined Z(V). The Harder--Narasimhan theorem states that every coherent sheaf on the curve admits a filtration by semistable vector bundles and skyscraper sheaves. It follows that every complex of coherent sheaves in the derived category of the curve admits a filtration by shifts of semistable bundles and skyscraper sheaves. A Bridgeland stability condition on a triangulated category is an abstraction of the above phenomenon. It picks out a class of semistable objects which provide building blocks for the category, and provides numerical criteria for determining when a given object is semistable. Formal definition A stability condition on a triangulated category \mathcal{D} is a pair (Z,\mathcal{P}) where the slicing \mathcal{P} is a collection of full additive subcategories \mathcal{P}(\varphi)\subset\mathcal{D} for each \varphi\in \mathbb{R}, and the central charge Z\colon \mathsf{K}_0(\mathcal{D})\to \mathbb{C} is a group homomorphism from the Grothendieck group of \mathcal{D} to the additive group of the complex numbers, satisfying the following condition: ;Shift-periodicity : An object A lies in \mathcal{P}(\varphi) if and only if A[1]\in \mathcal{P}(\varphi+1). ;Right-orthogonality : If A\in \mathcal{P}(\varphi_1), B\in \mathcal{P}(\varphi_2), and \varphi_1 > \varphi_2, then \operatorname{Hom}(A,B)=0. ;Compatibility of slicing and central charge :For every \varphi\in\mathbb{R} and non-zero object A\in\mathcal{P}(\varphi), there is a positive real number m(A) (called the mass of A) such thatZ(A)=m(A)\cdot e^{i\pi\varphi}. ;Existence of Harder--Narasimhan filtrations: For every object E\in \mathcal{D} there exists a finite sequence of real numbers \varphi_1>\varphi_2>\cdots>\varphi_n, and objects A_i\in \mathcal{P}(\varphi_i) for each i sitting in the sequence of exact triangles below. For a stability condition \sigma=(Z,\mathcal{P}) and a real number \varphi\in\mathbb{R}, a non-zero object A\in\mathcal{P}(\varphi) is said to be \sigma-semistable of phase \varphi (or simply semistable). If it additionally has no proper non-zero sub-objects in \mathcal{P}(\varphi), it is said to be stable. The support property was introduced by Kontsevich and Soibelman, as an abstraction of the observation that when the metric on a three-dimensional Calabi-Yau manifold approaches the large volume limit, the integrals of harmonic forms ("norms") on special Lagrangian submanifolds ("stable objects") is uniformly bounded by their volumes ("central charge") . Many authors include the support property in the definition of a stability condition, calling those without the support property pre-stability conditions instead. Stability conditions on curves For a smooth projective curve C, the function Z(-)=-\operatorname{deg}(-)+i\cdot \operatorname{rank}(-) extends to a well-defined central charge \mathsf{K}_0(\mathsf{D}^bC)\to \mathbb{C} on the derived category of coherent sheaves on C. Declaring \mathcal{P}(1) to be the full subcategory of torsion sheaves, \mathcal{P}(\varphi) for \varphi\in (0,1) to be the full additive subcategory generated by semistable vector bundles W with slope \mu(W)=-\tan(\varphi), and extending to remaining \varphi\in \mathbb{R} by shift-periodicity defines a slicing on \mathsf{D}^b C by the Harder--Narasimhan theorem. The pair (Z,\mathcal{P}) is a Bridgeland stability condition that has the support property. More generally, the construction can be repeated with any function Z_w(-)=-\operatorname{deg}(-)+w\cdot \operatorname{rank}(-)for a number w in the complex upper half-plane \mathbb{H}, each such pair (Z_w,\mathcal{P}_w) is a stability condition on \mathsf{D}^bC. == On abelian categories ==

A stability function on an abelian category \mathcal{A} is a group homomorphism Z\colon \mathsf{K}_0(\mathcal{A})\to \mathbb{C} such that for each non-zero object E\in \mathcal{A}, the complex number Z(E) lies in the semi-closed upper half plane \mathbb{R}_{. The phase of E is the real number \varphi(E)=\tfrac{1}{\pi}\operatorname{arg}Z(E)\in (0,1]. The object E is semi-stable (resp. stable) with respect to Z if for every proper non-zero sub-object F\subset E, the inequality \varphi(F)\leq \varphi(E) (resp. \varphi(F)) holds. The stability function Z has the Harder--Narasimhan property if every non-zero object E\in \mathcal{A} admits a filtration 0=E_0\subset E_1\subset \cdots \subset E_n=E such that each factor F_i=E_i/E_{i-1} is semistable and their phases satisfy \varphi(F_1)>\varphi(F_2)>\ldots >\varphi(F_n). Bridgeland showed that a stability condition (Z,\mathcal{P}) on a triangulated category \mathcal{D} is equivalent to the data of the heart of a bounded t-structure \mathcal{A}\subset\mathcal{D}, and a stability function Z\colon \mathsf{K}_0(\mathcal{A})\to \mathbb{C} with the Harder--Narasimhan property. There is a natural isomorphism of Grothendieck groups \mathsf{K}_0(\mathcal{A})\cong \mathsf{K}_0(\mathcal{D}) that turns the stability function into a central charge, and defining \mathcal{P}(\varphi)=\left\langle E\in \mathcal{A}\;|\; E\text{ semi-stable with }\varphi(E)=\varphi \right\ranglefor \varphi\in (0,1] uniquely specifies a slicing on \mathcal{D}. Conversely if (Z,\mathcal{P}) is a stability condition, then the subcategory \mathcal{P}(0,1]is the heart of a bounded t-structure on which Z gives a stability function with Harder--Narasimhan property. == The stability manifold ==

For an essentially small triangulated category \mathcal{D} with a fixed surjection \lambda\colon \mathsf{K}_0(\mathcal{D})\to \Lambda onto a free abelian group of finite rank, there is a complex manifold \operatorname{Stab}(\mathcal{D}) called the stability manifold whose points parametrise stability conditions (Z,\mathcal{P}) on \mathcal{D} for which the central charge Z factors through \lambda and has the support property with respect to this factorisation. A geometric stability condition on \mathcal{D} is a numerical stability condition (Z,\mathcal{P}) for which all skyscraper sheaves \mathcal{O}_p at closed points p\in Xare semistable and have the same phase, that is, there is a real number \varphi such that \{\mathcal{O}_p\;|\; p\in X\}\subset \mathcal{P}(\varphi). == Examples ==

Elliptic curves There is an analysis by Bridgeland for the case of elliptic curves. He finds there is an equivalence\text{Stab}(X)/\text{Aut}(X) \cong \text{GL}^+(2,\mathbb{R})/\text{SL}(2,\mathbb{Z})where \text{Stab}(X) is the set of stability conditions and \text{Aut}(X) is the set of autoequivalences of the derived category D^b(X). ==References==