Configurations in abelian categories. IV: Invariants and changing stability conditions
This is the final paper in a series of four papers on configurations in abelian categories by \textit{D. Joyce} [Adv. Math. 203, 194--255 (2006; Zbl 1102.14009), Adv. Math. 210, 635--706 (2007; Zbl 1119.14005), Adv. Math. 215, 153--219 (2007; Zbl 1134.14007)]. In this paper the author studies the variation of the moduli spaces \(Obj_{ss}^{\alpha}(\tau)\) with change in the (weak) stability condition from \((\tau, T,\leq)\) to \((\tilde{\tau}, \tilde{T},\leq)\). This variation is expressed by an identity in \(CF(Obj_{\mathcal A})\) (the algebra of constructible functions on \(Obj_{\mathcal A}\)) giving \(\delta^{\alpha}_{ss}(\tilde{\tau})\) in terms of \(\delta^{\beta_i}_{ss}(\tau)\) where, for \(X\in A\) with class \(\alpha\), the \(\beta_i\) are the classes of the \(\tau\)-semistable successive quotients in the Harder-Narasimhan filtration of \(X\). It is deduced that the algebras \(H^{pa}_{\tau}, \cdots, \overline{H}^{to}_{\tau}\) are indepedent of \((\tau,T,\leq)\). In the second part of the paper the author studies systems of invariants of \({\mathcal A}\) and \((\tau,T,\leq)\), with values in \({\mathbb Q}\)-algebras, which count semistable objects and configurations in \({\mathcal A}\). The transformation laws for these invariants under the change of stability conditions are obtained. Some additive and multiplicative identities among the invariants are established under certain conditions on the Ext groups of objects of \({\mathcal A}\). In the special cases when \({\mathcal A}= \mathrm{mod}-{\mathbb K}{\mathbb Q}\), the category of representations of quivers, or \({\mathcal A}=\mathrm{coh} P\), the category of coherent sheaves on a smooth curve \(P\), these invariants are computed explicitly recovering the results of Reineke and Harder-Narasimhan, Atiyah-Bott. For \(P\) a smooth surface with \(K_P^{-1}\) nef or a Calabi-Yau \(3\)-fold, invariants with special multiplicative transformation laws are found. For surfaces \(P\) with \(c_1(P)=0\), such as a \(K3\)-surfaces, invariants are found which are independent of the choice of Gieseker stability condition \((\tau, T,\leq)\). Connection of these invariants with Donaldson invariants of surfaces and Donaldson-Thomas invariants of Calabi-Yau 3-folds is investigated.
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