An obstruction to the existence of constant scalar curvature Kähler metrics

Details An obstruction to the existence of constant scalar curvature Kähler metrics An obstruction to the existence of constant scalar curvature Kähler metrics

2004-12-29

arxiv.org/abs/math/0412518v4

Journal of Differential Geometry 72, 429--466. 2006.

Mathematics

Differential Geometry

Submitted version incoorporating referee's corrections. Added notion of analytic K-stability following conversations with A. A

Scientific paper

We prove that polarised manifolds that admit a constant scalar curvature K\"ahler (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 K\"ahler 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 K\"ahler 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 K\"ahler 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.

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