Scalar curvature and Futaki invariant of Kähler metrics with cone singularities along a divisor

Abstract

We study the scalar curvature of Kähler metrics that have cone singularities along a divisor, with a particular focus on certain specific classes of such metrics that enjoy some curvature estimates. Our main result is that, on the projective completion of a pluricanonical bundle over a product of Kähler-Einstein Fano manifolds with the second Betti number 1, momentum-constructed constant scalar curvature Kähler metrics with cone singularities along the ∞-section exist if and only if the log Futaki invariant vanishes on the fibrewise C * -action, giving a supporting evidence to the log version of the Yau-Tian-Donaldson conjecture for general polarisations. We also show that, for these classes of conically singular metrics, the scalar curvature can be defined on the whole manifold as a current, so that we can compute the log Futaki invariant with respect to them. Finally, we prove some partial invariance results for them. n i=2 √ −1dz i ∧ dz i around D, with coordinates (z 1 , . . . , z n ) as above. The above definition is more restrictive than this usual definition, but will include all the cases that we shall treat in this paper (cf. Definition 1.10).Remark 1.3. We can regard a conically singular metric ω sing as a (1, 1)-current on X, and hence can make sense of its cohomology class [ω sing ] ∈ H 2 (X, R). 2Kähler-Einstein metrics with cone singularities along a divisor, studied initially in [27,37,47,50], attracted renewed interest since the foundational work of Donaldson [21] on the linear theory of Kähler-Einstein metrics with cone singularities along a divisor. Since then, there has already been a huge accumulation of research on such metrics.We now recall the log K-stability, which was introduced by Donaldson [21] and played a crucially important role in proving the Yau-Tian-Donaldson conjecture (Conjecture 2.6) for Fano manifolds; see Remark 2.16. We first recall (cf. Theorem 2.10) that the notion of K-stability can be regarded as an "algebro-geometric generalisation" of the vanishing of the Futaki invariant

Construction of constant scalar curvature Kähler cone metrics

“…(i)(ii). Hashimoto provides essentially the equivalence for a very particular bundle and . Remark that the considered bundle can be made parabolic polystable using the techniques .…”

“…From the point of view of existence, the case of curves has been studied by McOwen, Troyanov and Luo–Tian in the late 80s. In higher dimension, Hashimoto has recently obtained momentum‐constructed cscK cone metrics on the projective completion of a pluricanonical line bundle over a product of Kähler–Einstein Fano manifolds. This enabled him to give first evidence of the log Yau–Tian–Donaldson conjecture.…”

Uniqueness of constant scalar curvature Kähler metrics with cone singularities. I: reductivity

“…Unlike Kähler-Einstein case, there are several different definitions of singular cscK metrics with cone like singularities along a divisor ( [7], [16], [15]). In the work [16], it provided two ways to investigate this problem.…”

On uniform log $K$-stability for constant scalar curvature Kähler cone metrics

“…The condition (2.14) is not vacuous, and satisfied e.g. for the case considered in [35,Theorem 1.7]. The above lemma shows that, when there is a nontrivial holomorphic vector field v that preserves D the cone angle is uniquely determined by the vanishing of the log Futaki invariant (as long as v is "generic" in the sense that it satisfies (2.14)).…”

“…The hypothesis on the automorphism group is indeed necessary; see [35,Theorem 1.7 and Remark 1.8] and also Proposition 2.20.…”