An obstruction to the existence of constant scalar curvature Kähler metrics

Ross, Julius; Thomas, Richard

J. Differential Geom., Tome 72 (2006) no. 1, p. 429-466 / Harvested from Project Euclid

Résumé

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.

Publié le : 2006-03-14
Classification:

@article{1143593746,
     author = {Ross, Julius and Thomas, Richard},
     title = {An obstruction to the existence of constant scalar curvature K\"ahler metrics},
     journal = {J. Differential Geom.},
     volume = {72},
     number = {1},
     year = {2006},
     pages = { 429-466},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1143593746}
}

Ross, Julius; Thomas, Richard. An obstruction to the existence of constant scalar curvature Kähler metrics. J. Differential Geom., Tome 72 (2006) no. 1, pp.  429-466. http://gdmltest.u-ga.fr/item/1143593746/