Conformal foliations, K\"ahler twists and the Weinstein construction

TL;DR

This paper classifies K"ahler structures with special foliations, introduces new constructions using Weinstein's method, and produces novel examples of harmonic morphisms, balanced metrics, and scalar curvature metrics.

Contribution

It provides a classification of K"ahler structures with geodesic homothetic foliations and introduces new constructions via Weinstein's method related to Swann's twists.

Findings

New classes of holomorphic harmonic morphisms with arbitrary fiber dimensions.

Non-K"ahler balanced metrics conformal to K"ahler metrics.

Examples of non-Einstein constant scalar curvature K"ahler metrics.

Abstract

We classify both local and global K\"ahler structures admitting totally geodesic homothetic foliations with complex leaves. The main building blocks are related to Swann's twists and are obtained by applying Weinstein's method of constructing symplectic bundles to K\"ahler data. As a byproduct we obtain new classes of: holomorphic harmonic morphisms with fibres of arbitrary dimension from compact K\"ahler manifolds; non-K\"ahler balanced metrics conformal to K\"ahler ones (but compatible with different complex structures). Some classes of non-Einstein constant scalar curvature K\"ahler metrics are also obtained in this way.