An obstruction to the existence of constant scalar curvature K

By J. Ross, Richard P. Thomas. 2004

171 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7dde8d9e5399d91c5ee19dca4389b3ccb0f0c6fa

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Citations in the corpus

The J-flow and stability (2013, methodology): “We can compute the invariant Fα(χV,κ) using calculations from [14], and from this we obtain the following.”
Greatest lower bounds on the Ricci curvature of Fano manifolds (2009, background): “Stoppa gives an obstruction to solving this equation, generalising the slope stability obstruction to the existence of cscK metrics due to Ross-Thomas [20].”
Blowing up Kähler manifolds with constant scalar curvature, II (2005, unknown): “Ross-Thomas ([ 24 ], Example 5:30) have proved that the classes [!FS] "2PD[E1] "4 (PD[E2] +PD[E3])”
Existence of approximate Hermitian-Einstein structures on semi-stable bundles (2010, methodology): “The analogue of the Donaldson functional is in this case the Mabuchi K-energy [19], and several analogues of Mumford-Takemoto stability have also been introduced, including Chow-Mumford stability, K-stability (Tian [28], Donaldson [11]), uniform K-stability (Szekelyhidi [27]), slope-stability (Ross-”
Filtrations and test-configurations (2011, background): “The corresponding sequence of test-configurations is simply deformation to the normal cone of the point (0 : 1), with smaller and smaller parameters as k → ∞ (see Ross–Thomas [23]).”
A generalization of the Ross--Thomas slope theory (2009, result): “r bundles by Mumford and Takemoto, and systematically studied stabilities of varieties with Thomas. As its applications, many examples most of which are even smooth, are proved to be unstable [Ro06], [RT06], [RT07], [PR09]. The concept of slope has been extended to the orbifold case [RT09]. We shoul”
Hamiltonian 2-forms in Kähler geometry, III extremal metrics and stability (2005, methodology): “ dimS≥ 6, we are only able to show that the extremal polynomial is positive on (−1,1) ∩Q. Before ourwork, it wasbelieved that K-polystability implies slopeK-polystability in general, but the proof in [40, 41] only shows that it implies slope K-semistability, the gap being closely related to the issu”
A primer on Seshadri constants (2008, background): “Interest in slope stability stems in part from the fact that it gives a concrete obstruction to other geometric conditions { for instance it is implied by the existence of constant scalar curvature K ahler metrics (see [ RosTho06 ]).”
Birationally rigid hypersurfaces (2006, unknown): “ Mori ﬁber space in its birational equivalence class, and Bir(X) = Aut(X). According to a recent theorem of Odaka and Okada [OO], it also implies that Xis slope stable in the sense of Ross and Thomas [RT]. Theorem A has roots going back to the work of Fano on the birational geometry of threefolds [F”
Kähler metrics with constant weighted scalar curvature and weighted K‐stability (2018, background): “normal cone with respect to the infinity section S∞ ⊂ X, see [4, 73] but the Kähler class AQ on XQ defines a polarization only for rational values of z0.”