On the topology of geometric and rational orbits for algebraic group actions over valued fields
Speaker: Đào Phương Bắc

Time: 9:30 - 11:30, April 14, 2021

Venue: Room 612, A6, Institute of Mathematics, VAST

Abstract: In this talk, we study the relationship between Zariski and relative closedness for actions of (smooth) algebraic groups defined over valued (mainly local) fields of any characteristic. In particular, we use some recent basic results regarding the completely reducible subgroups and cocharacter-closedness due to Bate-Herpel-Roehrle-Tange and Uchiyama to construct some actions of simple algebraic groups $G$ of the types $D_{4}, E_{6}, E_{7}, E_{8}, G_{2}$ on an affine variety defined over a local function field $k$, and $v in V(k)$ such that the geometric orbit $G.v$ is Zariski closed although the corresponding relative orbit $G(k).v$ is not closed in the topology induced from $k$. Besides, we show that this phenomenon does not appear when we consider the action of nilpotent groups defined over an admissible valued (e.g., local) field. In fact, we show that the class of nilpotent groups is optimal in some sense. This is joint work with Vu Tuan Hien.

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