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Register a new userStability conditions for Slodowy slices and real variations of stability
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We provide examples of an explicit submanifold in Bridgeland stabilities space of a local Calabi-Yau, and propose a new variant of definition of stabilities on a triangulated category, which we call a real variation of stability conditions. We discuss its relation to Bridgelands definition; the main theorem provides an illustration of such a relation. We also state a conjecture by the second author and Okounkov relating this structure to quantum cohomology of symplectic resolutions and establish its validity in some special cases. More precisely, let X be the standard resolution of a transversal slice to an adjoint nilpotent orbit of a simple Lie algebra over C. An action of the affine braid group on the derived category of coherent sheaves on X and a collection of t-structures on this category permuted by the action have been constructed in arXiv:1101.3702 and arXiv:1001.2562 respectively. In this note we show that the t-structures come from points in a certain connected submanifold in the space of Bridgeland stability conditions. The submanifold is a covering of a submanifold in the dual space to the Grothendieck group, and the affine braid group acts by deck transformations. In the special case when dim (X)=2 a similar (in fact, stronger) result was obtained in arXiv:math/0508257.
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In this article, we treat stability conditions in the sense of King, Bridgeland and Bayer in a single framework. Following King, we begin with weight functions on a triangulated category, and consider increasingly specialised configurations of triangulated categories, t-structures and stability functions that give equivalent categories of stable objects. Along the way, we recover existing results in representation theory and algebraic geometry, and prove a series of new results on elliptic surfaces, including correspondence theorems for Bridgeland stability conditions and polynomial stability conditions, local finiteness and boundedness for mini-walls for Bridgeland stability conditions, isomorphisms between moduli of 1-dimensional twisted Gieseker semistable sheaves and 2-dimensional Bridgeland semistable objects, the preservation of geometric Bridgeland stability by autoequivalences on elliptic surfaces of nonzero Kodaira dimension, and solutions to Gepner equations on elliptic surfaces.
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