Abstract
We prove that constant scalar curvature Kähler (cscK) manifolds with transcendental cohomology class are K-semistable, naturally generalising the situation for polarised manifolds. Relying on a recent result by R. Berman, T. Darvas and C. Lu regarding properness of the K-energy, it moreover follows that cscK manifolds with discrete automorphism group are uniformly K-stable. As a main step of the proof we establish, in the general Kähler setting, a formula relating the (generalised) Donaldson–Futaki invariant to the asymptotic slope of the K-energy along weak geodesic rays.
Highlights
In this paper we are interested in questions of stability for constant scalar curvature Kähler manifolds with transcendental1 cohomology class
X, such questions are closely related to the Yau–Tian–Donaldson (YTD) conjecture [27,49,54]: A polarised algebraic manifold (X, L) is K-polystable if and only if the polarisation class c1(L) admits a Kähler metric of constant scalar curvature. This conjecture was recently confirmed in the Fano case, i.e. when L = −K X, cf. [16,17,18,52]. In this important special case, a constant scalar curvature Kähler (cscK) metric is nothing but a Kähler–Einstein metric
Several partial results had been obtained by Donaldson [28] and Stoppa [46], both assuming that c1(L) contains a cscK metric
Summary
In this paper we are interested in questions of stability for constant scalar curvature Kähler (cscK) manifolds with transcendental cohomology class. By analogy to the above papers, our proof relies on establishing such formulas valid for transcendental classes (see Theorems B and C), in particular relating the asymptotic slope of the K-energy along weak geodesic rays to a natural generalisation of the Donaldson–Futaki invariant. The above confirms one direction of the Yau–Tian–Donaldson conjecture, here referring to its natural generalisation to the case of arbitrary compact Kähler manifolds with discrete automorphism group, see Sect. 5.2
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