# K\"ahler manifolds with geodesic holomorphic gradients

## TL;DR

This paper classifies compact Kähler manifolds with special holomorphic gradient vector fields, showing they are all biholomorphic to bundles of complex projective spaces, thus revealing their geometric structure.

## Contribution

It provides a classification of such manifolds with geodesic holomorphic gradients under an integrability condition, identifying their biholomorphic equivalence to projective space bundles.

## Key findings

- Manifolds admit nontrivial real-holomorphic geodesic gradient vector fields.
- All such manifolds are biholomorphic to bundles of complex projective spaces.
- The classification relies on the integrability condition of the vector fields.

## Abstract

A vector field on a Riemannian manifold is called geodesic if its integral curves are reparametrized geodesics. We classify compact K\"ahler manifolds admitting nontrivial real-holomorphic geodesic gradient vector fields that satisfy an additional integrability condition. They are all biholomorphic to bundles of complex projective spaces.