The International Center for Mathematical Sciences – Sofia (ICMS-Sofia)


ICMS Geometry Seminar

15.11.2023, 16:00, Sofia time

ICMS-Sofia, Room 403 and via Zoom

Topics in non-archimedean analytic geometry (I)

Jiachang Xu, IMI-BAS

In 1960’s, Tate initiated the theory of rigid analytic variety as a foundation of geometry over non-archimedean fields (Qp , Cp , k ((t)) for any field k , etc.). Raynaud’s methods are based on formal algebraic geometry, which considers the rigid analytic geometry induced from a geometry of model, more precisely, Raynaud showed that the category of rigid analytic varieties is equivalent to the category of formal models over valuation ring up to certain admissible blowing-ups; this method enables us to study rigid analytic varieties by birational algebraic geometry. In the last few decades, Berkovich’s theory of k -analytic space has extended classic rigid geometry. Since k -analytic space has good topological properties, and the Berkovich analytification of algebraic varieties could be also dealt with via the geometry of the model and it has a strong connection with tropical geometry, this allows us to use the combinatorial techniques to study algebraic varieties. Our lectures will mainly discuss the contents of Berkovich space and its application in other aspects of mathematics, we plan to go over the basic properties of Berkovich space in both algebraic and topological ways for the first lecture.