On arithmetic varieties. II
[For part I see Lie groups Represent., Proc. Summer Sch. Bolyai János math. Soc., Budapest 1971, 151-217 (1975; Zbl 0308.14007.] - Let G be a simple algebraic group (over \({\mathbb{Q}})\) and K a maximal compact subgroup of the set \(G_{{\mathbb{R}}}\) of real points of G such that \(D=K\backslash G_{{\mathbb{R}}}\) has a \(G_{{\mathbb{R}}}\)-invariant complex structure and is further of dimension \(>1\). For a torsion-free arithmetic subgroup \(\Gamma\) in \(G_{{\mathbb{R}}}\), \(X=X_{\Gamma}:=D/\Gamma\) is a smooth complex manifold which is a so-called arithmetic variety and there exists a (''multicanonical'') projective imbedding of X; the closure \(\bar X\) of X is a normal variety (algebraic, by Chow's theorem, with \(\bar X_ a\) denoting the corresponding \({\mathbb{C}}\)-variety) and further \(Y:=\bar X\backslash X\) is a projective variety. For any automorphism \(\sigma\) of \({\mathbb{C}}\), let \(\bar X_ a^{\sigma}\) (resp. \(\bar Y_ a^{\sigma})\) be the variety obtained from \(\bar X_ a\) (resp. \(\bar Y_ a)\) by base change and \(\bar X^{\sigma}\) the variety of \({\mathbb{C}}\)-points of \(\bar X_ a^{\sigma}\). If \(X^{\sigma}\subset \bar X^{\sigma}\) is the open variety corresponding to \(X_ a^{\sigma}:=\bar X_ a^{\sigma}- \bar Y_ a^{\sigma},\) then the author's main theorem states that for any \(\sigma \in Aut {\mathbb{C}},\) the variety \(X^{\sigma}\) is arithmetic. For \(G=Sp(2n,.)\), the main theorem was known to be true on interpreting X as a moduli space for polarised abelian varieties and for ''classical'' groups G (i.e. with an admissible morphism of G into \(Sp(2n,.)),\) as a result of Deligne's work. For anisotropic groups G, the author proved the main theorem earlier and in the present case, (wherein an interpretation in terms of moduli is unknown) the proof follows the same lines as in the anisotropic case.
scientific article; zbMATH DE number 3863368
scientific article