Configurations in abelian categories. IV: Invariants and changing stability conditions
type: Item
instance of: scholarly article
title: Configurations in abelian categories. IV: Invariants and changing stability conditions
zbMATH Open document ID: 1134.14008
published in: Advances in Mathematics
publication date: 2007-11-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0410268
review text: 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 \(K3\)-…
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 18E10
Mathematics Subject Classification ID: 14A20
zbMATH DE Number: 5213655
zbMATH Keywords: abelian category
zbMATH Keywords: configuration
zbMATH Keywords: stability condition
zbMATH Keywords: moduli space
zbMATH Keywords: constructible function
Wikidata QID: Q56990968
author: Dominic Joyce
reviewed by: Usha N. Bhosle
MaRDI profile type: Publication
OpenAlex ID: W2084157173
arXiv ID: math/0410268
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DOI: 10.1016/J.AIM.2007.06.011
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