Fano deformation rigidity of rational homogeneous spaces of submaximal Picard numbers

Abstract

We study the question whether rational homogeneous spaces are rigid under Fano deformation. In other words, given any smooth connected family \(\pi :{{\mathcal {X}}}\rightarrow {{\mathcal {Z}}}\) of Fano manifolds, if one fiber is biholomorphic to a rational homogeneous space S, is \(\pi \) an S-fibration? The cases of Picard number one were answered by Hwang and Mok. The manifold \({{\mathbb {F}}}(1, Q^5)\) 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 \(\rho (G/P)\le \mathrm{rank}(G)\). Weber and Wiśniewski proved that the rational homogeneous spaces G/P with \(\rho (G/P)=\mathrm{rank}(G)\) (i.e. complete flag manifolds) are rigid under Fano deformation. In this paper, we show that the rational homogeneous spaces G/P with \(\rho (G/P)=\mathrm{rank}(G)-1\) are rigid under Fano deformation, provided that G is a simple algebraic group of type ADE, and G/P is not biholomorphic to \({{\mathbb {F}}}(1, 2, {{\mathbb {P}}}^3)\) or \({{\mathbb {F}}}(1, 2, Q^6)\). We also show that \({{\mathbb {F}}}(1, 2, {{\mathbb {P}}}^3)\) has a unique Fano degeneration, which is explicitly constructed. Furthermore, the structure of possible Fano degenerations of \({{\mathbb {F}}}(1, 2, Q^6)\) is also described explicitly. Our main result is obtained by applying the theory of Cartan connections and symbol algebras.