T-structures on some local Calabi-Yau varieties

Let $Z$ be a Fano variety satisfying the condition that the rank of the Grothendieck group of $Z$ is one more than the dimension of $Z$. Let $\omega_Z$ denote the total space of the canonical line bundle of $Z$, considered as a non-compact Calabi-Yau variety. We use the theory of exceptional collections to describe t-structures on the derived category of coherent sheaves on $\omega_Z$. The combinatorics of these t-structures is determined by a natural action of an affine braid group, closely related to the well-known action of the Artin braid group on the set of exceptional collections on $Z$.Comment: 30 page

Stability conditions on triangulated categories

This paper introduces the notion of a stability condition on a triangulated category. The motivation comes from the study of Dirichlet branes in string theory, and especially from M.R. Douglas's notion of $\Pi$-stability. From a mathematical point of view, the most interesting feature of the definition is that the set of stability conditions \Stab(\T) on a fixed category \T has a natural topology, thus defining a new invariant of triangulated categories. After setting up the necessary definitions I prove a deformation result which shows that the space \Stab(\T) with its natural topology is a manifold, possibly infinite-dimensional.Comment: A minor change in terminology (centered slope function becomes stability function). The result on stability conditions on curves of positive genus is removed since E. Macri found a much better proof in math/0411613. A false statement pointed out by S. Okada has also been removed. To appear in Annals of Math