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In this question it is asked if every flag manifold can be given the structure of a Kähler manifold. In the first answer it is written

Flag manifolds exhaust all compact homogeneous Kähler manifolds corresponding to a compact connected semi-simple Lie group

What is the standard reference for this result?

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1 Answer 1

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Flag manifolds $G/C(S)$ even exhaust homogeneous symplectic manifolds of $G$: Borel-Weil (1954, Thm 1). Also restated with fewer details in (1954, Thm 1).

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  • $\begingroup$ Sorry, I'm confused by "even homogeneous symplectic". Every symplectic manifold is automatically even dimensional - is there some meaning to "odd dimensional homogeneous symplectic"? $\endgroup$
    – Fofi Konstantopoulou
    Aug 20, 2019 at 14:01
  • $\begingroup$ Better now? (“even” = adverb qualifying exhaust, not adjective qualifying manifolds.) $\endgroup$
    – Francois Ziegler
    Aug 20, 2019 at 14:09
  • $\begingroup$ Ah, I see :) Thanks for clearing this up! $\endgroup$
    – Fofi Konstantopoulou
    Aug 20, 2019 at 14:10

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