Fano deformation rigidity of rational homogeneous spaces of submaximal Picard numbers

Abstract

© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.We study the question whether rational homogeneous spaces are rigid under Fano deformation. In other words, given any smooth connected family π: X→ Z of Fano manifolds, if one fiber is biholomorphic to a rational homogeneous space S, is π an S-fibration? The cases of Picard number one were answered by Hwang and Mok. The manifold F(1 , Q5) is the unique rational homogeneous space of Picard number one that is not rigid under Fano deformation, and a Fano degeneration of it is constructed by Pasquier and Perrin. For higher Picard number cases, one notices that the Picard number of a rational homogeneous space G/P satisfies ρ(G/ P) ≤ rank (G). Weber and Wiśniewski proved that the rational homogeneous spaces G/P with ρ(G/ P) = rank (G) (i.e. complete flag manifolds) are rigid under Fano deformation. In this paper, we show that the rational homogeneous spaces G/P with ρ(G/ P) = rank (G) - 1 are rigid under Fano deformation, provided that G is a simple algebraic group of type ADE, and G/P is not biholomorphic to F(1 , 2 , P3) or F(1 , 2 , Q6). We also show that F(1 , 2 , P3) has a unique Fano degeneration, which is explicitly constructed. Furthermore, the structure of possible Fano degenerations of F(1 , 2 , Q6) is also described explicitly. Our main result is obtained by applying the theory of Cartan connections and symbol algebras.