Holomorphic Parabolic Geometries and Calabi-Yau Manifolds

TL;DR

This paper proves that Calabi-Yau manifolds admit only homogeneous complex parabolic geometries, specifically on complex tori, and provides a classification of such geometries on homogeneous compact Kähler manifolds.

Contribution

It establishes a classification result showing the exclusivity of homogeneous geometries on Calabi-Yau manifolds and characterizes complex parabolic geometries on homogeneous compact Kähler manifolds.

Findings

Calabi-Yau manifolds only admit homogeneous geometries on complex tori.

Complete classification of complex parabolic geometries on homogeneous compact Kähler manifolds.

Identification of the limited types of geometries compatible with Calabi-Yau structures.

Abstract

We prove that the only complex parabolic geometries on Calabi-Yau manifolds are the homogeneous geometries on complex tori. We also classify the complex parabolic geometries on homogeneous compact K\"ahler manifolds.