On arithmetic varieties. II
type: Item
instance of: scholarly article
title: On arithmetic varieties. II
zbMATH Open document ID: 0543.14030
DOI: 10.1007/BF02760617
published in: Israel Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: [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.
Mathematics Subject Classification ID: 14L35
Mathematics Subject Classification ID: 11E57
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 14L40
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 14D20
zbMATH DE Number: 3863368
zbMATH Keywords: arithmeticity of variety
author: David Kazhdan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3863368