Fano deformation rigidity of rational homogeneous spaces of submaximal Picard numbers

We study the question whether rational homogeneous spaces are rigid under Fano deformation. In other words, given any smooth connected family f:X -> Zof Fano manifolds, if one fiber is biholomorphic to a rational homogeneous space S, whether is f an S-fibration? The cases of Picard number one were studied in a series of papers by J.-M. Hwang and N. Mok. For higher Picard number cases, we notice that the Picard number of a rational homogeneous space G/P is less or equal to the rank of G. Recently A. Weber and J. A. Wisniewski proved that rational homogeneous spaces G/P with Picard numbers equal to the rank of G (i.e. complete flag manifolds) are rigid under Fano deformation. In this paper we show that the rational homogeneous space G/P is rigid under Fano deformation, providing that G is a simple algebraic group of type ADE, the Picard number equal to rank(G)-1 and G/P is not biholomorphic to F(1, 2, P^3) or F(1, 2, Q^6). The variety F(1, 2, P^3) is the set of flags of projective lines and planes in P^3, and F(1, 2, Q^6) is the set of flags of projective lines and planes in 6-dimensional smooth quadric hypersurface. We show that F(1, 2, P^3) have a unique Fano degeneration, which is explicitly constructed. The structure of possible Fano degeneration of F(1, 2, Q^6) is also described explicitly. To prove our rigidity result, we firstly show that the Fano deformation rigidity of a homogeneous space of type ADE can be implied by that property of suitable homogeneous submanifolds. Then we complete the proof via the study of Fano deformation rigidity of rational homogeneous spaces of small Picard numbers. As a byproduct, we also show the Fano deformation rigidity of other manifolds such as F(0, 1, ..., k_1, k_2, k_2+1, ..., n-1, P^n) and F(0, 1, ..., k_1, k_2, k_2+1, ..., n, Q^{2n+2}) with 0 <= k_1 < k_2 <= n-1.