An obstruction to the existence of constant scalar curvature Kähler metrics

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dc.creatorRoss, J.
dc.creatorThomas, R. P.
dc.date2004-12-29
dc.date2005-12-05
dc.date.accessioned2026-07-07T06:39:13Z
dc.date.available2026-07-07T06:39:13Z
dc.descriptionWe 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 $μ(Z)\leμ(X)$ 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.
dc.descriptionSubmitted version incoorporating referee's corrections. Added notion of analytic K-stability following conversations with A. Apsotolov and D. Calderbank
dc.identifierhttps://arxiv.org/abs/math/0412518
dc.identifierhttp://arxiv.org/abs/math/0412518
dc.identifierJournal of Differential Geometry 72, 429--466. 2006.
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/100997
dc.subjectDifferential Geometry
dc.subjectAlgebraic Geometry
dc.subject32Q15; 53C21; 14L24
dc.titleAn obstruction to the existence of constant scalar curvature Kähler metrics
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