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Braid groups and Kleinian singularities. (English) Zbl 1264.14026
Consider a finite group \(G\subset \mathrm{SL}(2,\mathbb{C})\), the surface \(\mathbb{C}^2/G\) with a unique singular point \(0\) and the minimal resolution \(\pi: X\rightarrow \mathbb{C}^2/G\). It is well-known that the exceptional divisor \(E=\pi^{-1}(0)\) consists of a tree of \((-2)\)-curves whose dual graph is a Dynkin diagram of type ADE. The structure sheaf \(S_i=\mathcal{O}_{E_i}\) of any irreducible component \(E_i\) of \(E\) is known to be a \(2\)-spherical object, that is, for any \(i\) we have \(S_i\otimes \omega_X\cong S_i\) and \(\text{Hom}(S_i,S_i[k])=\mathbb{C}\) if \(k=0,2\) and \(0\) otherwise. Any spherical object gives rise to a spherical twist, an autoequivalence of the bounded derived category of coherent sheaves \(D^b(X)\) on \(X\), and it was shown by P. Seidel and R. Thomas [Duke Math. J. 108, No. 1, 37–108 (2001; Zbl 1092.14025)] that the spherical twists associated to the \(S_i\) satisfy the braid relations. In type A they also showed that this action of the braid group is faithful. The purpose of the paper under review is to prove the faithfulness in types ADE.
To do this, the Garside structure on braid groups is used in an essential way. Background concerning this and further generalities on braid groups are presented in Section 2. In Section 3 the authors recall some facts about spherical twists and prove their main result. The idea is to show that a braid group element is completely determined by the action of the corresponding twist on the direct sum of the \(S_i\). In the last section an application to spaces of stability conditions is presented. Namely, T. Bridgeland has shown in [Int. Math. Res. Not. 2009, No. 21, 4142–4157 (2009; Zbl 1228.14012)] that a connected component of the stability manifold of a certain subcategory \(\mathcal{D}\) of \(D^b(X)\) is a covering of the space of regular orbits of the Weyl group corresponding to the singularity type. He also proved that if the braid group action is faithful, then the covering is in fact the universal one. Hence, the results of the paper under review show that we have a universal covering not only in type A but in types ADE.

MSC:
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14B05 Singularities in algebraic geometry
20F36 Braid groups; Artin groups
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