Stability conditions and Kleinian singularities

By T. Bridgeland. 2005

105 citations recorded by the source corpus.

We describe (connected components of) the spaces of stability conditions on certain triangulated categories associated to Dynkin diagrams. These categories can be defined either algebraically via module categories of preprojective algebras, or geometrically via coherent sheaves on resolutions of Kleinian singularities. The resulting spaces of stability conditions are covering spaces of regular subsets of the corresponding complexified Cartan algebras.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dac42514263d8c36f4c767d3d05712cc0bd8469b

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Stability Conditions on An-Singularities (2010, methodology): “Hence we have the canonical group homomorphism π1(ĥ reg/Ŵ ) = B(1)n × Z→ Br(D)× Z. Bridgeland also shows that this homomorphism coincides with ρ̃.”
Relative singularity categories I: Auslander resolutions (2012, unknown): “This follows [19] from a faithfulness result for the braid group actions generated by spherical twists (see [84] for type A and [18] for general Dynkin types).”
Inducing stability conditions (2007, background): “Via the McKay correspondence, the category Db(X) is equivalent to Db(Ab 2), the derived category of the preprojective algebra associated to the aﬃne Dynkin diagram Ab2: ◦ @@ @@ @@ @ ~~ ~~ ~~ ~ • • By [9], the natural faithful functor Db(A2) −→ Db(Ab2) factorizes as Db(A2) −→i D → Db(Ab 2). Hence, th”
Flat surfaces and stability structures (2014, unknown): “more, Stab(C) comes with an action of Aut(C) and the universal cover of GL+(2;R). Thanks to the eorts of a number of people, the structure of Stab(C) is understood in many particular cases, see e.g. [6, 10, 11, 24, 29, 13]. Moduli spaces of very similar nature appear in the theory of complex curves ”
Affine braid group actions on derived categories of Springer resolutions (2011, background): “2 is related to spherical twists and mirror symmetry, and has been extensively studied by several mathematicians (see [57, 17, 37, 16]).”
The space of stability conditions on the local projective plane (2009, background): “fundamental difference to the case of Calabi-Yau 2-categories (as studied in [Bri08, Tho08, Bri05a , IUU06, HMS08]).”
Moduli stacks and invariants of semistable objects on K3 surfaces (2007, background): “3 is satisfied when X is a K3 surface or an Abelian surface, and Stab(X) is the connected component described in [7].”
Braid groups and Kleinian singularities (2009, methodology): “Bridgeland further shows that the image of ρ : B → Aut(D) gives the full group of deck transformations for the covering p, and therefore that if ρ is injective, p is in fact a universal cover.”