K\"ahlerity of Einstein four-manifolds

TL;DR

This paper establishes conditions under which Einstein four-manifolds are either anti-self-dual or K"ahler-Einstein, linking curvature eigenvalues to complex geometric structures.

Contribution

It proves a new curvature condition involving the eigenvalues of the self-dual Weyl tensor that characterizes K"ahler-Einstein or anti-self-dual four-manifolds.

Findings

Einstein four-manifolds with $ abla W^+=0$ are either anti-self-dual or K"ahler-Einstein.

The eigenvalue condition $oxed{ ext{if }\, ext{eigenvalue}_2 \, ext{of } W^+ \, ext{ satisfies } \, oxed{rac{S}{12}}}$ implies K"ahler-Einstein.

The result generalizes previous classifications by incorporating eigenvalue bounds.

Abstract

We prove that a closed oriented Einstein four-manifold is either anti-self-dual or (after passing to a double Riemann cover if necessary) K\"ahler-Einstein, provided that $\lambda_2 \geq -\frac{S}{12}$, where $\lambda_2$ is the middle eigenvalue of the self-dual Weyl tensor $W^+$ and $S$ is the scalar curvature. The same conclusion holds for closed oriented four-manifolds with $\delta W^+=0$.