The complex geometry of two exceptional flag manifolds

Abstract

We discuss the complex geometry of two complex five-dimensional Kähler manifolds which are homogeneous under the exceptional Lie group G 2 . 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 G 2 -invariant almost complex structures on these manifolds. The full flag manifold G 2 /T 2 is discussed briefly in Section 4 below. 1 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 [6,17,32]. 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 [14] and Yau [36] that on the manifold underlying complex projective space the standard structure is the unique Kählerian complex structure. Since [14], such rigidity results have been proved for a few other manifolds, for example for the odd-dimensional quadrics by Brieskorn [5]. Like the result of Hirzebruch and Kodaira, many of these extensions depend on the fact that they consider manifolds with very simple cohomology algebras. We refer the reader to [24,23,8] for accounts of some refined results in the spirit of [14]. As explained in [8] and the references given there, any compact Kähler manifold with the integral cohomology ring of CP 5 is biholomorphic to it. The manifolds Q and Z show that this fomulation is sharp. They are simply connected compact oriented 10manifolds with the same homology and cohomology groups as CP 5 , but with different and distinct ring structures on cohomology 2 .Brieskorn's result [5] shows that the manifold Q has a unique Kählerian complex structure, without any assumption about it being homogeneous or Fano. We prove below the analogous statement for the manifold Z.