Formal functions on homogeneous spaces
The author generalizes one of his previous theorems: Suppose \(X=G/H\) is an algebraic variety (over a field), homogeneous under a connected algebraic group G. Suppose further that \(Y\subseteq X\) is a subvariety generating G, that is (if Y contains the origin) the connected component of 1 of the preimage of Y in G generates G. Under these assumptions any formal meromorphic function on the formal completion of X along Y is algebraic. The proof uses rigid analytic methods, but in fact Grothendieck's algebraisation for formal schemes suffices. The key is the behaviour under formal completion of the K/k-span of the variety: If \(K\supseteq k\) is a field extension, X a variety defined over k, and \(Y\subseteq X_ K\) a K-subvariety, its K/k-span is defined to be the smallest k-subvariety of X containing Y. In a previous result [Ann. Math., II. Ser. 89, 391-403 (1969; Zbl 0184.465)] the author considered complex analytic meromorphic functions.
scientific article; zbMATH DE number 3995877