# Einstein Metrics, Harmonic Forms, and Conformally Kaehler Geometry

## TL;DR

This paper explores the geometry of Einstein 4-manifolds with specific curvature conditions related to harmonic forms, extending previous classifications and highlighting the importance of transversality in these geometric structures.

## Contribution

It generalizes earlier classifications of Einstein 4-manifolds by analyzing cases with non-negative self-dual Weyl curvature and emphasizes the critical role of transversality conditions.

## Key findings

- Classification of Einstein 4-manifolds with non-negative self-dual Weyl curvature
- Identification of the importance of transversality in geometric properties
- Different phenomena emerge when transversality condition is dropped

## Abstract

The author has elsewhere given a complete classification of those compact oriented Einstein 4-manifolds on which the self-dual Weyl curvature is everywhere positive in the direction of some self-dual harmonic 2-form. In this article, similar results are obtained when the self-dual Weyl curvature is everywhere non-negative in the direction of a self-dual harmonic 2-form that is transverse to the zero section of the bundle of self-dual 2-forms. However, this transversality condition plays an essential role in the story; dropping it leads one into wildly different territory where entirely different phenomena predominate.