Characterization of standard embeddings between complex Grassmannians by means of varieties of minimal rational tangents
Motivated by the relations between function theory on bounded symmetric domains and the geometric theory of Fano manifolds, the author gives a new proof of a theorem of Tsai in the case of irreducible symmetric domains of type I. This theorem states that a proper holomorphic map between an irreducible bounded symmetric domain of rank \(r>1\) and another bounded domain of rank \(r' \leq r\) is necessarily totally geodesic. The proof is in two steps. First, the author embeds the domain in the Grassmann compactification and uses boundary values to prove that the map \( f\) transforms rank 1 vectors to rank 1 vectors, on a nonempty open set. Then, thanks to the study of geometric structures by means of rational curves which provides a new proof of a result of Neretin, he proves that \(f\) is actually a standard embedding between complex Grassmann manifolds.
scientific article; zbMATH DE number 5323346
scientific article