Log homogeneous compactifications of some classical groups (Q2339180)

Let \(k\) be an algebraically closed field, and let \(G\) be a connected reductive \(k\)-group. In [\textit{D. Luna} and \textit{Th. Vust}, Comment. Math. Helv. 58, 186--245 (1983; Zbl 0545.14010)] one finds a classification theory of embeddings of \(\Omega,\) a homogeneous space under the action of \(G\), in the case where \(k\) has characteristic zero. An embedding here is a normal irreducible variety with a \(G\)-action which contains \(\Omega\) as a dense orbit. The focus of the work under review is on compactifications of these embeddings (which are complete embeddings) of certain types, particularly log homogeneous compactifications. It is known that if char \(k=0\) then \(\Omega\) admits a log homogeneous compactification if and only if a Borel subgroup of \(G\) has a dense orbit in \(\Omega\), i.e. if and only if \(G\) is spherical\(.\) In this paper, this result is adapted to the case where char \(k=p>0,\) wherein \(\Omega\) has a log homogeneous orbit if and only if \(G\) is separably spherical (i.e., \(\Omega\) is spherical and \(G\) has a Borel subgroup whose open orbit in \(\Omega\) is separable). This is then related to the regular and colorless compactifications. Also, the equivariant compactification is studied. In particular, the construction of such a compactification is made explicit.