Einstein Metrics, Complex Surfaces, and Symplectic 4-Manifolds

TL;DR

This paper characterizes which smooth compact 4-manifolds with complex or symplectic structures admit Einstein metrics with non-negative Einstein constant, providing a complete classification in these special cases.

Contribution

It offers a complete classification of 4-manifolds with complex or symplectic structures that admit Einstein metrics with non-negative Einstein constant.

Findings

Complete classification for complex 4-manifolds admitting Einstein metrics.

Complete classification for symplectic 4-manifolds admitting Einstein metrics.

Identification of specific topological and geometric conditions for existence.

Abstract

Which smooth compact 4-manifolds admit an Einstein metric with non-negative Einstein constant? A complete answer is provided in the special case of 4-manifolds that also happen to admit either a complex structure or a symplectic structure.