Convexity of the weighted Mabuchi functional and the uniqueness of weighted extremal metrics

Abstract

We prove the uniqueness, up to a pull-back by an element of a suitable subgroup of complex automorphisms, of the weighted extremal Kähler metrics on a compact Kähler manifold introduced in our previous work [26]. This extends a result by and Chen-Paun-Zeng [15] in the extremal Kähler case. Furthermore, we show that a weighted extremal Kähler metric is a global minimum of a suitable weighted version of the modified Mabuchi energy, thus extending our results from [26] from the polarized to the Kähler case. This implies a suitable notion of weighted K-semistability of a Kähler manifold admitting a weighted extremal Kähler metric.

Weighted K-stability of polarized varieties and extremality of Sasaki manifolds

“…As Ť is simultaneously central and maximal, it follows that the identity component of Aut Ť r (M, J) is abelian and equal to ŤC . The result now follows from the uniqueness result in [41], established in the more general context of (v, w)-extremal Kähler metrics and building on an argument in [16], which implies that any two Ť-invariant ( Ǩ, κ)-extremal Kähler metrics in ζ are isometric by an element in the connected component of the identity of the group Aut Ť r (M, J). With this understood, for any Theorem 4.12.…”

“…This expression is independent of the Ť-invariant Kähler metric Ω ∈ A according to [40]; furthermore, it is not hard to see that the numerical invariant does not depend on the chosen momentum polytope P for (M, ζ, Ť): it merely depends upon the data ( Ǩ, κ, Ť) so we shall denote it by F ext ( Ǩ,κ) (M , A ). In this more general Kähler setting, [41,Theorem 2] implies that if ζ admits a ( Ǩ, κ)-extremal Kähler metric, then F ext Ǩ,κ (M , A ) ≥ 0 for any smooth Ť-equivariant test configuration with reduced central fibre associated to (M, J, ζ, Ť), and we can extend Conjecture 5.8 to a relative ( Ǩ, κ)-weighted K-stability statement.…”

“…For proving the conditions (P1)-(P7) of [21], we notice that: (P1) follows from [35,Theorem 6.7] and the fact that the (relative) Mabuchi energy M K differs from the functional K in [35] by an affine-linear function on bounded geodesics, see [9,Prop. 10] or [41] for a direct argument with respect to M X K on the regular quotient; (P2) follows from [35,Theorem 6.6]; (P3) is the regularity result [34,Theorem 5.4] which can be extended from the CSC to the extremal case by the argument in [33]. (P4) and (P5) are established in Lemmas 4.7 and 4.9 above.…”

“…One can extend the conclusions of Theorems 4.12 and 5.5 to smooth polarized test configurations with reduced central fibre, which are T-equivariant with respect to the torus T Aut(M, L) generated by the Sasaki-Reeb vector fields X, K, and the CR vector field K ext of Definition B.1. To this end, one needs to consider instead of G = T C , the action of the centralizer G T of T in Aut(M, L) (which also acts on the spaces R X (V, J) and R K (V, J)), and use the uniqueness of T-invariant K-extremal metrics modulo G T (which is established [41]) in order to obtain the properness of M Ǩ,κ with respect to G T .…”

“…This allows us to use the extension of M K to the d 1 -completion and the regularity of its minimizers (obtained in [34,35]) to show the properness of M K (see Theorem 4.12) and hence of M Ǩ,κ . A key result for this to work is the transitivity of the action of T C on the space of T-invariant K-extremal Kähler metrics in 2πc 1 (L), which has been obtained in [41] using the approach in [16]. Once the T C -properness of M Ǩ,κ is established, the proof of Theorem 1 essentially follows from the arguments in [8,52].…”