Seminar on Arithmetic Geometry

Speaker: Helene Sigloch (Uni-Freiburg)

Date, time & place: December 10, 2015, 10:15 am, WSC-N-U-3.05

Title: Vector Bundles over Rigid Analytic Varieties

Abstract: Classification of vector bundles has always been a popular problem all over geometry. Grauert (1957) classified holomorphic vector bundles on Stein spaces up to isomorphism by homotopy classes of maps into the infinite Grassmannian. Morel (2012) classsified algebraic vector bundles on smooth affine varieties up to isomorphism by $\mathbb{A}^1$-homotopy classes into the infinite Grassmannian. I want to do the same for vector bundles over rigid analytic varieties.
Using a trick of Schlichting's, Asok-Hoyois-Wendt managed to considerably generalise Morel's theorem and establish a similar theorem for principal bundles under certain algebraic groups. Transferring their technique to the rigid analytic setting, I could prove that isomorphism classes of line bundles correspond to rigid analytic $\mathbb{A}^1$-homotopy classes to a classifying space.
In the talk I will give a short introduction into rigid analytic varieties and affinoid algebras. I will explain the question and state the known results, barely touching abstract homotopy theory and instead giving some explicit computations.