An obstruction to the existence of constant scalar curvature Kähler metrics

Abstract

We prove that polarised manifolds that admit a constant scalar curvature Kähler (cscK) metric satisfy a condition we call slope semistability. That is, we define the slope μ for a projective manifold and for each of its subschemes, and show that if X is cscK then μ(Z) ≤ μ(X) for all subschemes Z. This gives many examples of manifolds with Kähler classes which do not admit cscK metrics, such as del Pezzo surfaces and projective bundles. If P(E) → B is a projective bundle which admits a cscK metric in a rational Kähler class with sufficiently small fibres, then E is a slope semistable bundle (and B is a slope semistable polarised manifold). The same is true for all rational Kähler classes if the base B is a curve. We also show that the slope inequality holds automatically for smooth curves, canonically polarised and Calabi-Yau manifolds, and manifolds with c₁(X) > 0 and L close to the canonical polarisation.