Holomorphic geometric structures on Kaehler-Einstein manifolds

TL;DR

This paper classifies compact Kaehler manifolds with specific Chern class conditions that admit certain holomorphic geometric structures, linking them to well-known geometric varieties and symmetric spaces.

Contribution

It provides a classification of Kaehler-Einstein manifolds with holomorphic parabolic and cominiscule geometries, connecting geometric structures to algebraic and symmetric space classifications.

Findings

Manifolds with nonnegative first Chern class and holomorphic parabolic geometries are flat bundles over complex tori.

Manifolds with negative first Chern class and holomorphic cominiscule geometries are locally Hermitian symmetric varieties.

The results unify geometric structure theory with the classification of special Kaehler manifolds.

Abstract

We prove that the compact Kaehler manifolds with first Chern class nonnegative that admit holomorphic parabolic geometries are the flat bundles of rational homogeneous varieties over complex tori. We also prove that the compact Kaehler manifolds with negative first Chern class that admit holomorphic cominiscule geometries are the locally Hermitian symmetric varieties.