The complex geometry of two exceptional flag manifolds
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The complex geometry of two exceptional flag manifolds D. Kotschick1 · D. K. Thung2 Received: 4 October 2019 / Accepted: 15 February 2020 © The Author(s) 2020
Abstract We discuss the complex geometry of two complex five-dimensional Kähler manifolds which are homogeneous under the exceptional Lie group G2. For one of these manifolds, rigidity of the complex structure among all Kählerian complex structures was proved by Brieskorn; for the other one, we prove it here. We relate the Kähler assumption in Brieskorn’s theorem to the question of existence of a complex structure on the six-dimensional sphere, and we compute the Chern numbers of all G2-invariant almost complex structures on these manifolds. Keywords Flag manifolds · Chern numbers · Kähler geometry Mathematics Subject Classification Primary 14M15 · 53C26 · 53C30; Secondary 14J45 · 32Q60 · 57R20
1 Introduction In this paper, we study the complex geometry of the two homogeneous spaces Q and Z appearing in the diagram of G2-invariant fibrations displayed in Fig. 1. They are both (co-) adjoint orbits of G2 , of the form G2 ∕U(2) , for two non-conjugate embeddings U(2) ↪ G2 . These subgroups are maximally parabolic, and the quotients are examples of exceptional partial flag manifolds.1 The manifold Z is the Salamon [30] twistor space of the exceptional Wolf [37] space M = G2 ∕SO(4) considered as a quaternionic Kähler manifold of positive scalar curvature. As such it has the structure of a smooth Fano variety, and it carries a holomorphic contact 1
The full flag manifold G2 ∕T 2 is discussed briefly in Sect. 4.
The second author is supported by the German Research Foundation (DFG) as part of RTG 1670. * D. Kotschick [email protected] D. K. Thung daniel.thung@uni‑hamburg.de 1
Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstr. 39, 80333 München, Germany
2
Department of Mathematics, Universität Hamburg, Bundesstr. 55, 20146 Hamburg, Germany
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D. Kotschick, D. K. Thung
Fig. 1 Diagram of fibrations between G2-homogeneous spaces; cf. [31, p. 164] and [35]
structure. The other quotient of G2 by U(2) is denoted by Q because it is diffeomorphic to a smooth quadric hypersurface in ℂP6 . Thus, it also carries the structure of a smooth Fano variety. Indeed, the complex structures are G2-invariant and there is a unique invariant Kähler–Einstein metric of positive scalar curvature in both cases. The distinction between U(2)− and U(2)+ is best described in terms of octonions, as in [7, 18, 35]. Without getting involved in the details, one can always distinguish Q and Z by remembering that the isotropy representation of Q splits into three irreducible summands, whereas the isotropy representation of Z has only two summands.
1.1 Rigidity of standard complex structures It is a classical result of Hirzebruch–Kodaira [15] and Yau [39] that on the manifold underlying complex projective space the standard structure is the unique Kählerian complex structure. Since [15], such rigidity results have been proved for a
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