On Fano complete intersections in rational homogeneous varieties
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Mathematische Zeitschrift
On Fano complete intersections in rational homogeneous varieties Chenyu Bai1 · Baohua Fu1,2 · Laurent Manivel3 Received: 30 August 2018 / Accepted: 24 May 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Complete intersections inside rational homogeneous varieties provide interesting examples of Fano manifolds. For example, if X = ∩ri=1 Di ⊂ G/P is a smooth complete intersection ∗ of r ample divisors such that K G/P ⊗ OG/P (− i Di ) 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.
1 Introduction We work within the category of complex projective varieties, unless stated otherwise. Rational homogeneous varieties are among the simplest algebraic varieties, and a better understanding of them is always a motivation for the development of algebraic geometry. For example, the solution by Mori of the Hartshorne conjecture characterizes projective spaces by the ampleness of its tangent bundle, which is a milestone of the minimal model program. A more recent conjecture of Campana–Peternell claims that rational homogeneous varieties are the only smooth rational varieties with nef tangent bundles, which is still far from being resolved. Complete intersections in rational homogeneous varieties provide many interesting examples of Fano varieties. It is expected by Hartshorne that all smooth subvarieties in Pn of small codimension are complete intersections, which is again far from being resolved. In this
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Baohua Fu [email protected] Chenyu Bai [email protected] Laurent Manivel [email protected]
1
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China
2
MCM, AMSS, Chinese Academy of Sciences, 55 ZhongGuanCun East Road, Beijing 100190, China
3
Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, UPS IMT, 31062 Toulouse Cedex 9, France
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C. Bai et al.
paper, we will study two geometrical properties of Fano complete intersections in rational homogeneous varieties: local rigidity and quasi-homogeneity. Recall that a smooth projective variety X is said locally rigid if for any smooth deformation X → B with X0 X , we have Xt X for t in a small (analytic) neighborhood of 0. By Kodaira-Spencer deformation theory, if H 2 (X , TX ) = 0, then X is locally rigid if and only if H 1 (X , TX ) = 0. For any rational homogeneous variety G/P, it is shown in [4, Theorem VII] that H i (G/P, TG/P ) = 0 for all i ≥ 1, hence they are locally rigid. In [3], the local rigidity is proven for Fano regular G-varieties. The case of two-orbits varieties of Picard number one is studied in [17]. Let G/P be a rational homogeneous variety with G simple and X = ∩ri=1 Di ⊂ G/P ∗ a smooth irreducible complete intersection of r ample divisors. We assume that K G/P ⊗ OG/P (− i Di ) is ample, which implies that X is Fano. When G/P is of Pi
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