On Fano complete intersections in rational homogeneous varieties

Complete intersections inside rational homogeneous varieties provide interesting examples of Fano manifolds. For example, if X=∩i=1rDi⊂G/P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X = \cap _{i=1}^r D_i \subset G/P$$\end{document} is a smooth complete intersection of r ample divisors such that KG/P∗⊗OG/P(-∑iDi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{G/P}^* \otimes {\mathcal O}_{G/P}(-\sum _i D_i)$$\end{document} is ample, then X is Fano. We first classify these Fano complete intersections which are locally rigid. It turns out that most of them are hyperplane sections. We then classify general hyperplane sections which are quasi-homogeneous.