On a Theorem of Wang for Complex Homogeneous Manifolds

Summary · qwen2.5:32b

The article provides a new Lie theoretic proof for Wang's theorem regarding the sufficiency conditions for a compact homogeneous manifold $G/H$ to admit a $G$-invariant complex structure, relying solely on root space decomposition properties. This contrasts with Ni and Wallach’s recent work that employs more specialized Lie algebra objects like Borel subalgebras and Iwasawa decompositions.

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Exploring a new proof of Wang's theorem on complex homogeneous manifolds in mathematical differential geometry.

Excerpt

arXiv:2607.02193v2 Announce Type: replace Abstract: In \cite{Wang1954}, Wang proved (among other things) a sufficiency result for a compact homogeneous manifold $G/H$ to admit a $G$-invariant complex structure. In this note, we give a new Lie theoretic proof of Wang's theorem which relies on nothing more than the familiar properties of the root space decomposition of a compact Lie group. It should be noted that the recent work of Ni and Wallach \cite{NiWallach2025} also revisits the aforementioned theorem of Wang (and others) and offers new Lie theoretic proofs as well. However, the approach of \cite{NiWallach2025} relies on such objects as Borel subalgebras, parabolic subalgebras, and Iwasawa decomposition which may be somewhat less familiar to the working differential geometer.