tailieunhanh - Đề tài " 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 . Douglas’s work on Π-stability. From a mathematical point of view, the most interesting feature of the definition is that the set of stability conditions Stab(D) on a fixed category D has a natural topology, thus defining a new invariant of triangulated categories. In a separate article [6] I give a detailed description of this space of stability conditions in the case that D is the bounded derived category of coherent. | Annals of Mathematics Stability conditions on triangulated categories By Tom Bridgeland Annals of Mathematics 166 2007 317 345 Stability conditions on triangulated categories By Tom Bridgeland 1. Introduction 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 . Douglas s work on n-stability. From a mathematical point of view the most interesting feature of the definition is that the set of stability conditions Stab D on a fixed category D has a natural topology thus defining a new invariant of triangulated categories. In a separate article 6 I give a detailed description of this space of stability conditions in the case that D is the bounded derived category of coherent sheaves on a K3 surface. The present paper though is almost pure homological algebra. After setting up the necessary definitions I prove a deformation result which shows that the space Stab D with its natural topology is a manifold possibly infinite-dimensional. . Before going any further let me describe a simple example of a stability condition on a triangulated category. Let X be a nonsingular projective curve and let D X denote its bounded derived category of coherent sheaves. Recall 11 that any nonzero coherent sheaf E on X has a unique Harder-Narasimhan filtration 0 E0 c E1 c c En-1 c En E whose factors Ej Ej-1 are semistable sheaves with descending slope 1 deg rank. Torsion sheaves should be thought of as having slope rc and come first in the filtration. On the other hand given an object E E D X the truncations ơ j E associated to the standard t-structure on D X fit into triangles j1 E ------ V Aj ơ j E --------- V Aj 1 Ơ j 1 E 318 TOM BRIDGELAND which allow one to break up E into its shifted cohomology sheaves Aj Hj E j . Combining these two ideas one can concatenate the Harder-Narasimhan filtrations of the cohomology sheaves Hj E to obtain a kind of filtration