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This paper contains a description of one connected component of the space of stability conditions on the bounded derived category of coherent sheaves on a complex algebraic K3 surface.

Let G G be a finite group of automorphisms of a nonsingular three-dimensional complex variety M M , whose canonical bundle ω M \omega _M is locally trivial as a G G -sheaf. We prove that the Hilbert scheme Y = G Y = G - Hilb ⁡ M \operatorname {Hilb}M parametrising G G -clusters in M M is a crepant resolution of X = M / G X=M/G and that there is a derived equivalence (Fourier–Mukai transform) between coherent sheaves on Y Y and coherent 𝐺-sheaves on M M . This identifies the K theory of Y Y with the equivariant K theory of M M , and thus generalises the classical McKay correspondence. Some higher-dimensional extensions are possible.

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Abstract. We prove that moduli spaces of meromorphic quadratic differentials with simple zeroes on compact Riemann surfaces can be identified with spaces of stability conditions on a class of CY 3 triangulated categories defined using quivers with potential associated to triangulated surfaces. We relate the finite-length trajectories of such quadratic differentials to the stable objects of the corresponding stability condition.

We use Joyce’s theory of motivic Hall algebras to prove that reduced Donaldson-Thomas curve-counting invariants on Calabi-Yau threefolds coincide with stable pair invariants and that the generating functions for these invariants are Laurent expansions of rational functions.

Abstract. We describe quantum enveloping algebras of symmetric Kac-Moody Lie algebras via a finite field Hall algebra construction involving Z 2 -graded complexes of quiver representations.

AWe give a condition for an exact functor between triangulated categories to be an equivalence. Applications to Fourier-Mukai transforms are discussed. In particular, we obtain a large number of such transforms for K3 surfaces.