Hodge Splittings and Einstein 4-manifolds

Abstract

On an oriented $4$-manifold, we study pairs of Riemannian metrics $(g,h)$ for which the curvature tensor of $g$ preserves the Hodge splitting determined by $h$. This extends the Einstein condition in dimension four, which is recovered when $h$ is conformal to $g$. We prove that such pairs satisfy a generalized Hitchin-Thorpe inequality, which reduces to the classical one when $h$ is conformal to $g$. We then exhibit a pair $(g,h)$ on ${\#}_5{\mathbb{CP}}^2$, which violates Hitchin-Thorpe and hence admits no Einstein metric, thus showing that our condition is indeed broader than the Einstein condition.

25 pages; v3: major revision, with new relative Hitchin-Thorpe results and a Hitchin-Thorpe-violating example