On Fano complete intersections in rational homogeneous varieties (Q2182412)

Complete intersections in rational homogeneous varieties provide many interesting examples of Fano varieties. It is expected by Hartshorne that all smooth subvarieties \(\mathbb{P}^n\) of small codimension are complete intersections. In this paper, the authors studies two geometrical properties of Fano complete intersections in rational homogeneous varieties: local rigidity and quasi-homogeneity. \textit{R. Bott} [Ann. Math. (2) 66, 203--248 (1957; Zbl 0094.35701)] shows that \(H^i(G/P, T_{G/P}) = 0\) for all \(i \geq 1\) and any rational homogeneous variety \(G/P\), hence they \(G/P\) is locally rigid by Kodaira-Spencer deformation theory. In [\textit{F. Bien} and \textit{M. Brion}, Compos. Math. 104, No. 1, 1--26 (1996; Zbl 0910.14004)], the local rigidity is proven for Fano regular \(G\)-varieties. The case of two-orbits varieties of Picard number one is studied in [\textit{B. Pasquier} and \textit{N. Perrin}, Math. Z. 265, No. 3, 589--600 (2010; Zbl 1200.14097)]. Let \(G/P\) be a rational homogeneous variety with \(G\) simple and \(X =\cap D_i \subset G/P\) a smooth irreducible complete intersection of \(r\) ample divisors. Suppose that \(K^*_{G/P}\otimes \mathcal{O}_{G/P}(-\sum D_i)\) is ample, which implies that \(X\) is Fano. When \(G/P\) is of Picard number one, the converse holds, but in general this condition is stronger than the Fanoness of X. The main theorem of this paper classifies such \(X\) which are locally rigid. It uses the fact that \(H^i(X, T_X) = 0\) for all \(i \geq 2\) by Kodaira-Nakano vanishing theorem; so \(X\) is locally rigid if and only if \(H^1(X, T_X) = 0\). The authors explain their theorem using Vinberg's theory of parabolic prehomogeneous spaces [\textit{L. Manivel}, Rend. Semin. Mat., Univ. Politec. Torino 71, No. 1, 35--118 (2013; Zbl 1362.11099)]. Given a connected Dynkin diagram \(D\), with a special node satisfying some conditions, we obtain a simply connected simple Lie group \(G\) and a maximal parabolic subgroup \(P\) with the following property. There is an embedding of \(G/P\) in the projectivization \(\mathbb{P}V_P^*\) of a (dualized) fundamental representation, such that \(G \times GL_k\) acts on \(V_P \otimes \mathbb{C}^k\) with finitely many orbits. In particular \(G\) acts on \(Gr(k, V_P )\) with only finitely many orbits, and therefore there exists only a finite number of isomorphism types of codimension \(k\) linear sections of \(G/P\). In this situation, the local rigidity of the general section can be expected, and this is exactly what happens. It turns out that most of the varieties obtained in the main theorem are hyperplane sections. The authors classify general hyperplane sections which are quasi-homogeneous. An observation is that a general hyperplane section of \(G/P\) is quasi-homogeneous if and only if it is locally rigid but not a hyperplane section of \(Gr(3, 8)\). In general, there is no direct relation between the two properties.