On the Picard functor in formal-rigid geometry

In this thesis, we report three preprints [Li17a] [Li17b] and [HL17] the author wrote (the last one was written jointly with D. Hansen) during his pursuing of PhD at Columbia. We study smooth proper rigid varieties which admit formal models whose special fibers are projective. The main theorem asserts that the identity components of the associated rigid Picard varieties will automatically be proper. Consequently, we prove that non-archimedean Hopf varieties do not have a projective reduction. The proof of our main theorem uses the theory of moduli of semistable coherent sheaves. Combine known structure theorems for the relevant Picard varieties, together with recent advances in p-adic Hodge theory, We then prove several related results on the low-degree Hodge numbers of proper smooth rigid analytic varieties over p-adic fields.