On the topology of relative orbits for actions of algebraic groups over complete fields. (English) Zbl 1217.14031

Suppose \(k\) is a field which is complete with respect to a non-trivial valuation of real rank \(1\) (for example, \(k\) could be a \(\mathfrak{p}\)-adic field). Let \(G\) be a linear algebraic \(k\)-group acting on an affine \(k\)-variety \(V\). Suppose \(x \in V(k)\) is a closed \(k\)-point. The main aim of this paper is to investigate connections between the Zariski closedness of the \(G\)-orbit \(G\cdot x\) in \(V\) and the Hausdorff closedness of the so-called relative orbit \(G(k)\cdot x\) in \(V(k)\). The methods involve finding ways of equipping certain cohomology groups with a topology.
The paper contains a review of many of the existing results in the literature, in particular results by D. Birkes [Ann. Math. (2) 93, 459–475 (1971; Zbl 0198.35001)] and R. J. Bremigan [J. Reine Angew. Math. 453, 21–47 (1994; Zbl 0808.14040)], and also presents some new results. The authors state that proofs of their results will appear elsewhere.


14L24 Geometric invariant theory
14L30 Group actions on varieties or schemes (quotients)
20G15 Linear algebraic groups over arbitrary fields
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