Stability of Arakelov bundles over arithmetic curves and Bridgeland stability conditions on holomorphic triples

Abstract

Part I - Arakelov bundles over arithmetic curves. In Chapter I we compile the basics about Arakelov geometry that we briefly described above. We define Arakelov vector bundles on arithmetic curves and we explore the relationship of nefness and the tensor product problem as evidence of the pathologies of the Arakelov setting. Chapter II reproduces Behrend’s construction of complementary polyhedra for stability of group schemes and the later adaptation to Arakelov geometry by Harder and Stuhler. The main results of this part are contained in chapter III, where we define Arakelov principal bundles. We provide a notion of stability and prove that our definition agrees with all the previous constructions. Part II - Bridgeland stability conditions on holomorphic triples over curves. Chapter IV gathers basic facts about triangulated and derived categories. In Chapter V we introduce the general definition of Bridgeland stability conditions and explore few examples of constructions of stability conditions that are interesting for our constructions. Finally, chapter VI contains all our constructions of Bridgeland stability conditions on holomorphic triples over curves (join work with A. Rincon and A. Rueffer). First we describe the bounded derived category of holomorphic triples on curves TC as semiorthogonal decomposition of the bounded derived category of coherent sheaves on the curve and we construct the Serre functor STC . Next, we compare recollement and CP-gluing to construct hearts via semiorthogonal decompositions, by gluing hearts in the smaller categories and we compute the necessary numerical conditions for triples. Finally, we construct stability conditions on TC by gluing stability conditions from Stab(C). We study the Harder-Narasimhan and the support properties of glued stability conditions in general and for triples. The very last section shows the sketch of how we finally come up with the full description of the stability manifold Stab(TC).