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March 2006 An obstruction to the existence of constant scalar curvature Kähler metrics
Julius Ross, Richard Thomas
J. Differential Geom. 72(3): 429-466 (March 2006). DOI: 10.4310/jdg/1143593746


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.


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Julius Ross. Richard Thomas. "An obstruction to the existence of constant scalar curvature Kähler metrics." J. Differential Geom. 72 (3) 429 - 466, March 2006.


Published: March 2006
First available in Project Euclid: 28 March 2006

zbMATH: 1125.53057
MathSciNet: MR2219940
Digital Object Identifier: 10.4310/jdg/1143593746

Rights: Copyright © 2006 Lehigh University


Vol.72 • No. 3 • March 2006
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