Configurations in abelian categories. III: Stability conditions and identities
The author has a series of four long papers on configurations in an abelian category \(A\), this one being the third in the series [for the other parts, cf. Adv. Math. 203, 194--255 (2006; Zbl 1102.14009), Adv. Math. 210, 635--706 (2007; Zbl 1119.14005); Adv. Math. 217, No. 1, 125--204 (2008; Zbl 1134.14008)]. For a finite partially ordered set \((I,\preceq)\) in \(A\), an \((I,\preceq)\)-configuration is a finite collection of objects in \(I\) and morphisms between them satisfying certain conditions. Let \(K(A)\) be the Grothendieck group of \(A\) and \(C(A)\subset K(A)\) the collection of classes in \(K(A)\) of all non-zero objects in \(A\). Denote by \(Obj_A\) and \(Obj_A^{\alpha}\) respectively the moduli stacks of objects in \(A\) and of objects in \(A\) with class \(\alpha\) in \(K(A)\). Generalizing \textit{A. Rudakov}'s stability conditions [J. Algebra, 197, 231--245 (1997; Zbl 0893.18007)], the author defines a (weak) stability condition to be a triple \((\tau, T, \leq)\) where \((T,\leq)\) is a totally ordered set and \(\tau: C(A) \to T\) is a map with certain properties. Notions of \(\tau\)-(semi)stability, \(\tau\)-unstability for objects in \(A\) are introduced. For example, slope functions for quiver representations and Gieseker stability of coherent sheaves are stability conditions while \(\mu\)-stability and purity of coherent sheaves are examples of weak stability conditions. It is shown that if \((\tau, T, \leq)\) is a (permissible) stability condition, then each \(\tau\)-semistable object \(X\) has a (non-unique) filtration with unique \(\tau\)-stable factors (successive quotients) \(S_k\) satisfying \(\tau(S_k)= \tau(X)\). If \((\tau, T, \leq)\) is a weak stability condition, then each \(X\) has a unique Harder-Narasimhan filtration with \(\tau\)-semistable factors \(S_k\) such that \(\tau(S_1)>\cdots >\tau(S_n=X)\). The author forms moduli spaces \(Obj_{ss}^{\alpha}, Obj_{st}^{\alpha},Obj_{si}^{\alpha}\), of \(\tau\)-semistable, \(\tau\)-stable and \(\tau\)-semistable indecomposable objects with class \(\alpha\). All these spaces are constructible sets of \(Obj_A\), hence it is easier to study the variation of these spaces as \((\tau, T, \leq)\) varies. These being constructible subsets, the characteristic functions \(\delta^{\alpha}_{ss},\delta^{\alpha}_{st}(\tau), \delta^{\alpha}_{si}\) of these sets are constructible functions. It is also shown that associated configuration moduli spaces \(M_{ss}, M_{si}, M_{st}, M^b_{ss}, M^b_{si}, M^b_{st}(I,\preceq,k,\tau)\) are constructible in the moduli stack \(M(I,\preceq)\) of configurations in \(A\). Consequently their characteristic functions \(\delta_{ss}, \delta_{si}, \delta_{st},\delta^b_{ss},\delta^{b}_{si},\delta^{b}_{st}(I,\preceq,k,\tau)\) are constructible. Many identities relating these functions and their stack function analogues are proved. Interesting algebras \(H_{\tau}^{pa}, H_{\tau}^{to},{\overline H}_{\tau}^{pa}, {\overline H}_{\tau}^{to}\) of constructible and stack functions on \(Obj^{\alpha}_A\) are defined and studied.
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