On the nonexistence of $S^6$ type complex threefolds in any compact homogeneous complex manifolds with the compact lie group $G_2$ as the base manifold

Daniel Guan

Vol. 305 (2020), No. 2, 641–644

DOI: 10.2140/pjm.2020.305.641

Abstract

In a recent preprint Professor Etesi asked a question: could one find a complex three dimensional submanifold $S$ in a compact complex seven dimensional homogeneous space with the compact real $14$ dimensional Lie group $G_2$ as the base manifold, such that $S$ is diffeomorphic to the six dimensional sphere $S^6$? We apply a result of Tits on compact complex homogeneous space, or of H. C. Wang and Hano–Kobayashi on the classification of compact complex homogeneous manifolds with a compact reductive Lie group to give an answer to his question. In particular, we show that one could not obtain a complex structure of $S^6$ in his way.

Keywords

complex structure, six dimensional sphere, cohomology, invariant structure, complex torus bundles, Hermitian manifolds

Mathematical Subject Classification 2010

Primary: 53C15

Secondary: 57S25, 53C30, 22E99, 15A75

Milestones

Received: 31 May 2019

Revised: 28 August 2019

Accepted: 7 November 2019

Published: 29 April 2020