Bridgeland stability and complex geometry I.

Building on ideas  from string theory,  Tom Bridgeland has introduced a notion of stability for elements in an arbitrary triangulated category. This new version of stability generalizes the classical concept of stability of vector bundles on curves. 

One of the main observations of Bridgeland's theory is that his so-called stability conditions form a complex manifold under not too restrictive hypotheses.  In particular, specializing to the case of the derived category of a smooth projective variety, we learn that there is a host of new stability conditions that we can hope to use for purposes of projective geometry.

There is an extensive amount of recent work on the application of stability conditions to the theory of linear series.