Seminar: Non-archimedean geometry

Bogdan Zavyalov

Abstract: Let X be a smooth and proper variety over a local field K. Then the weight-monodromy conjecture predicts that the monodromy and weight filtrations coincide up to a shift. Recently, P. Scholze proved this conjecture for set-theoretic complete intersections inside the projective space using the theory of perfectoid spaces. Alternatively, one can formulate a (local) version of the weight-monodromy conjecture for the nearby cycles. We will give a precise formulation of this conjecture and prove it in some cases following the strategy of Scholze in the global case. This is joint work with David Hansen.