An obstruction to the existence of constant scalar curvature Kähler metrics

Abstract

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. 1

Asymptotic Chow polystability in Kähler geometry

“…Theorem 7.4 ([34], [35]). If we put w(r, k) = This result says that if e r ≤ 0 for all large r then F 1 (M, L) ≤ 0, namely that asymptotic Chow semistability implies K-semistability.…”

Unstable products of smooth curves

“…The notion of slope stability for polarised varieties was introduced in [16,17] as a necessary condition for K-stability. The general idea is that a non-generic subscheme of a polarised variety (X, L) with certain numerical properties will have a slope that is too small, and this forces (X, L) to be unstable.…”

“…Letting K be the canonical divisor of X, a simple application of the RiemannRoch theorem to calculate a 0 (x) and a 1 (x) yields ( [16] …”

“…We letã i (x) = a i − a i (x) and for 0 < c ≤ ǫ(Z, L) the slope of X and Z is defined to be ( [16] (3.1,3.14)),…”

“…To show that certain Kähler classes do not admit cscK metrics we shall use an obstruction for K-stability (and hence for cscK metrics) of Thomas and the author called slope stability [16,17]. The main results of this paper are the following: Theorem 3.7.…”