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HOLOMORPHIC SECTIONAL CURVATURE OF FLAG MANIFOLDS VIA KÄHLER REDUCTION

A mix of human-written and AI-generated textHuman understanding: all partsmath.DG — Differential Geometry

Contributed by Ching Hsien Lee ↗

Version 1 / Nov 07, 2026 / CC BY 4.0

Abstract

We give an alternative proof that every invariant Kähler metric on a compact generalized flag manifold has positive holomorphic sectional curvature. Invariant Kähler metrics on unitary partial flag manifolds are realized as reductions of Hermitian vector spaces at compact moment-map levels contained in spheres. A curvature identity for linear Kähler quotients gives an explicit positive lower bound. For an inclusion of compact connected Lie groups, we prove that the induced inclusion between adjoint orbits through the same Lie algebra element is holomorphic, isometric, and totally geodesic for the Kähler structures determined by an invariant inner product and its restriction. This reduces the general case to unitary flags. We also obtain a strict estimate for the second fundamental form of the anticanonical projective embedding

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Version history

  1. v1 · 9 pagesSubmitted by Ching Hsien LeeInitial depositCurrentNov 06, 2026