Rational points over homogeneous spaces and their compactifications
The main object of the paper under review is a homogeneous space \(X\) of an algebraic group \(G\) defined over a perfect field \(k\). The author's goal is to prove that if some \(k\)-variety birationally equivalent to \(X\) has a smooth rational point, then so does \(X\). In particular, one can deduce the existence of a rational point on \(X\) from the existence of a rational point on its smooth compactification. This result is a far-reaching generalization of a theorem by \textit{J.-L.~Colliot-Thélène} and \textit{J.-J.~Sansuc} [Duke Math. J. 54, 375--492 (1987; Zbl 0659.14028)] which concerned the case where \(G\) is an algebraic \(k\)-torus. The strategy of the proof is as follows. The author proves a stronger statement saying that the existence of a \(k\)-point follows from the existence of a \(k((t))\)-point. If \(X\) is a \(G\)-torsor, this statement is equivalent to the injectivity of the restriction map \(H^1(k,G)\to H^1(k((t)),G)\). To go over from torsors to arbitrary homogeneous spaces, one has, roughly speaking, to pass from nonabelian \(H^1\) to nonabelian \(H^2\). This is mainly what the author is doing. The main technical tool he uses is the notion of acyclic extension \(l/k\) (by definition, this means that the restriction map \(H^1(k,G)\to H^1(l,G)\) is bijective for all \(k\)-groups \(G\)). He notices that if \(l/k\) is acyclic, then the existence of a \(k\)-point on \(X\) follows from the existence of an \(l\)-point. It remains to construct an acyclic extension \(K_{\infty}\) of \(k\) containing \(k((t))\) (if the characteristic of \(k\) equals zero, it is enough to adjoin to \(k\) all roots of \(t\)). The proof can then be finished by a simple geometric argument allowing one, roughly, to move a smooth rational point along an integral curve. Apart from the main theorem, some of auxiliary results are of their own interest (as, for example, considerations around the restriction map for nonabelian \(H^2\)).
scientific article; zbMATH DE number 5118033