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Deformation rigidity of the 20-dimensional \(F_4\)-homogeneous space associated to a short root. (English) Zbl 1071.22012
Popov, Vladimir L. (ed.), Algebraic transformation groups and algebraic varieties. Proceedings of the conference on interesting algebraic varieties arising in algebraic transformation group theory, Vienna, Austria, October 22–26, 2001. Berlin: Springer (ISBN 3-540-20838-0/hbk). Encyclopaedia of Mathematical Sciences 132. Invariant Theory and Algebraic Transformation Groups 3, 37-58 (2004).
Let \(G\) be a complex simple Lie group and \(P\) a maximal parabolic subgroup; recall that the quotient \(G/P\) is a rational homogeneous space of Picard number 1. The authors work on the following conjecture: Let \(\pi\) be a smooth projective morphism from a complex manifold \(X\) to the unit disc, and assume that for each \(t\neq0\) the fiber \(\pi^{-1}(t)\) is biholomorphic to \(G/P\). Then \(\pi^{-1}(0)\) is also biholomorphic to \(G/P\).
If we denote by \(\mathfrak g\) the Lie algebra of \(G\) and by \(\alpha\) the simple root determining \(P\), then in the previous papers by the authors [Invent. Math. 131, No. 2, 393–418 (1998; Zbl 0902.32014); J. Reine Angew. Math. 486, 153–163 (1997; Zbl 0876.53030); Ann. Sci. Éc. Norm. Supér., IV. Sér. 35, No. 2, 173–184 (2002; Zbl 1008.32012)], the above conjecture was proved for all rational homogeneous spaces of Picard number 1 except for the following three cases: (1) type \((C_l,\alpha)\), where \(\alpha\) is a short root; (2) type \((F_4,\alpha_1)\); (3) type \((F_4,\alpha_2)\).
The purpose of the paper under review is to settle the case (3), i.e. of \((F_4,\alpha_2)\). The main difference against the proofs in the three papers mentioned above is that now the structure of the relevant symbol algebra cannot be recovered from the Serre presentation of the simple Lie algebra, but has to be gleaned directly from the projective geometry of the variety of minimal rational tangents. For this reason, a large portion of the paper is devoted to a study of this variety, which is of independent interest.
For the entire collection see [Zbl 1051.14003].

22E46 Semisimple Lie groups and their representations
22E10 General properties and structure of complex Lie groups
14M17 Homogeneous spaces and generalizations
53C35 Differential geometry of symmetric spaces
58H15 Deformations of general structures on manifolds