Higher Fano manifolds

In this paper we address Fano manifolds with positive higher Chern characters. They are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. For instance, they should be covered by higher dimensional rational varieties, and families of higher Fano manifolds over higher dimensional bases should admit meromorphic sections (modulo the Brauer obstruction). Aiming at finding new examples of higher Fano manifolds, we investigate positivity of higher Chern characters of rational homogeneous spaces. We determine which rational homogeneous spaces of Picard rank $1$ have positive second Chern character, and show that the only rational homogeneous spaces of Picard rank $1$ having positive second and third Chern characters are projective spaces and quadric hypersurfaces. We also classify Fano manifolds of large index having positive second and third Chern characters. We conclude by discussing conjectural characterizations of projective spaces and complete intersections in terms of these higher Fano conditions

Recent developments and open problems in linear series

In the week of October 3–9, 2010, the Mathematisches Forschungsinstitut at Oberwolfach hosted the Mini-Workshop “Linear Series on Algebraic Varieties.” These notes contain a variety of interesting problems which motivated the participants prior to the event, and examples, results and further problems which grew out of discussions during and shortly after the workshop. Many arguments presented here are scattered in the literature or constitute “folklore.” It was one of our aims to have a usable and easily accessible collection of examples and results