P-adic representations attached to vector bundles on smooth complete p-adic varieties

We discuss vector bundles with numerically stable reduction on smooth complete varieties over a p-adic number field and sketch the construction of associated p-adic representations of the geometric fundamental group. On projective varieties, such bundles are semistable with respect to every polarization and have vanishing Chern classes. One of the main problems in the construction consisted in getting rid of infinitely many obstruction classes. This is achieved by adapting a theory of Bhatt based on de Jongs's alteration method. One also needs control over numerically flat bundles on arbitrary singular varieties over finite fields. The singular Riemann Roch Theorem of Baum Fulton Macpherson is a key ingredient for this step. This is joint work with Annette Werner.