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 $\mu$ for a projective manifold and for each of its subschemes, and show that if $X$ is cscK then $\mu (Z)\le \mu (X)$ 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 $\PP(E)\to B$ 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 \emph{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 $c_1(X)<0$ and $L$ close to the canonical polarisation.

arXiv commentSubmitted version incoorporating referee's corrections. Added notion of analytic K-stability following conversations with A. Apsotolov and D. Calderbank

References in corpus (6)

• Numerical characterization of the Kähler cone of a compact Kähler manifold
• An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds, II
• Algebraic and Analytic K-Stability
• A remark on Kähler metrics of constant scalar curvature on ruled complex surfaces
• A study of the Hilbert-Mumford criterion for the stability of projective varieties
• Lower bounds on the Calabi functional

Cited by in corpus (8)

• CM Stability and the Generalized Futaki Invariant I
• Unstable products of smooth curves
• Notes on GIT and symplectic reduction for bundles and varieties
• Weighted projective embeddings, stability of orbifolds and constant scalar curvature Kähler metrics
• A remark on Kähler metrics of constant scalar curvature on ruled complex surfaces
• A study of the Hilbert-Mumford criterion for the stability of projective varieties
• On the limit of spectral measures associated to a test configuration
• Geometric Invariant Theory and Birational Geometry