Tom Bridgeland

The McKay correspondence as an equivalence of derived categories

Let G be a finite group of automorphisms of a nonsingular complex threefold M such that the canonical bundle omega_M is locally trivial as a G-sheaf. We prove that the Hilbert scheme Y=GHilb M parametrising G-clusters in M is a crepant resolution of X=M/G and that there is a derived equivalence (Fourier- Mukai transform) between coherent sheaves on Y and coherent G-sheaves on M. This identifies the K theory of Y with the equivariant K theory of M, and thus generalises the classical McKay correspondence. Some higher dimensional extensions are possible.

Stability conditions on

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.

K3

Abstract. We examine the extent to which a smooth minimal complex projective surface X is determined by its derived category of coherent sheaves (DX). To do this we find, for each such surface X, the set of surfaces Y for which there exists a Fourier-Mukai transform D

surfaces

(Y)\to

Complex surfaces with equivalent derived categories

D(X).

Hall algebras and curve-counting invariants

We use Joyces 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.

Fourier-Mukai transforms for K3 and elliptic fibrations

Given a non-singular variety with a K3 fibration π : X → S we construct dual fibrations π : Y → S by replacing each fibre Xs of π by a two-dimensional moduli space of stable sheaves on Xs. In certain cases we prove that the resulting scheme Y is a non-singular variety and construct an equivalence of derived categories of coherent sheaves Φ: D(Y ) → D(X). Our methods also apply to elliptic and abelian surface fibrations. As an application we use the equivalences Φ to relate moduli spaces of stable bundles on elliptic threefolds to Hilbert schemes of curves.

Stability Conditions and Kleinian Singularities

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.

Stability Conditions on a Non-Compact Calabi-Yau Threefold

We study the space of stability conditions Stab(X) on the non-compact Calabi-Yau threefold X which is the total space of the canonical bundle of