Abstract

There has been a lot of work done recently in the study of the bounded derived categories ofmoduli spaces of parabolic bundles on curves. Let C be a curve of genus at least two, and let L bea line bundle of odd degree on it. l was observed some time ago that the motive of the modulispace of rank 2 stable vector bundles on C with determinant L decomposes into motives ofsymmetric powers of this curve. Thus, it is natural to conjecture that the bounded derived categoryof this moduli space has an analogous semiorthogonal decomposition. The first results in thisdirection were obtained by Narasimhan and, independently, Kuznetsov and myself: an embeddingof the bounded derived category of the curve itself was constructed. This result initiated a series ofpapers which culminated in the whole decomposition being constructed (the last bit was done byTevelev who seemingly proved that the corresponding components generate the derived category)If we pass to genus zero, the theory of vector bundles is rather poor: every vector bundle splits intoa direct sum of line bundles; however, there is a reasonable substitute for moduli of stable bundlesin theses case moduli of parabolic bundles. We will give an overview of the higher genus story andask a similar guestion about the derived category of certain moduli of rank 2 parabolic bundles on aprojective line.