# Einstein four-manifolds with self-dual Weyl curvature of nonnegative determinant

## TL;DR

This paper characterizes simply connected Einstein four-manifolds with positive scalar curvature as conformally Kähler precisely when their self-dual Weyl curvature's determinant is positive, linking curvature conditions to complex structure.

## Contribution

It establishes a new equivalence condition connecting the positivity of the self-dual Weyl curvature determinant with conformally Kähler structures in Einstein four-manifolds.

## Key findings

- Conformally Kähler Einstein four-manifolds have positive self-dual Weyl curvature determinant.
- The determinant condition characterizes when such manifolds are conformally Kähler.
- Provides a curvature-based criterion for complex structure in Einstein four-manifolds.

## Abstract

We prove that simply connected Einstein four-manifolds of positive scalar curvature are conformally K\"ahler if and only if the determinant of the self-dual Weyl curvature is positive.