Constant $μ$-scalar curvature Kähler metric -- formulation and foundational results

Abstract

We introduce µ-scalar curvature for a Kähler metric with a moment map µ and start up a study on constant µ-scalar curvature Kähler metric as a generalization of both cscK metric and Kähler-Ricci soliton and as a continuity path to extremal metric. We study some fundamental constraints to the existence of constant µ-scalar curvature Kähler metric by investigating a volume functional as a generalization of Tian-Zhu's work, which is closely related to Perelman's W-functional. A new K-energy is studied as an approach to the uniqueness problem of constant µ-scalar curvature and as a prelude to new K-stability concept. Contents 18 4. µK-energy and µK-stability 25 4.1. µK-energy functional 25 4.2. A prelude to µK-stability 29 5. Openness of Kähler class and λ admitting µ-cscK 30 5.1. Regularity 30 5.2. Perturbation 30 References 32

Weighted K-stability and coercivity with applications to extremal Kahler and Sasaki metrics

“…Using (46), we conclude that MA X p (ϕ j ) → MA X p (ϕ) weakly as j → ∞. For an arbitrary weight function v ∈ C 0 (P), we take a sequence of polynomials p i of the above form converging to v in C 0 (P).…”

“…• v = 1 and w = const: this is the familiar cscK problem; • v = 1 and w = with un affine-linear function on t * : (1) then describes an extremal Kähler metric in the sense of Calabi [19]; • v = e , w = 2( + a)e where is an affine-linear function on t * and a is a constant correspond to the so-called µ-cscK [46], extending the notion of Kähler-Ricci solitons [64] defined when X is Fano and α = 2πc 1 (X).…”

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

“…Notice that if we take T = {1} and v = w ≡ 1, we obtain the much studied problem of the existence of cscK metric in α whereas taking T to be a maximal torus in Aut red (X) and v = w ≡ 1, our problem reduces to the famous Calabi problem of the existence of an extremal Kähler metric on (X, α). As we have noticed in [26], there is a number of other natural problems in Kähler geometry which can be reduced to the search of (v, w)-extremal Kähler metrics for special choices of T and the weight functions v and w, including the existence of conformally Kähler, Einstein-Maxwell metrics [3], the existence of extremal Sasaki metrics [1], the existence of Kähler-Ricci solitons [24,25,7], prescribing the scalar curvature on compact toric manifolds [20] and on semi-simple, rigid toric fibre bundles [2] as well as the recently introduced µ-cscK metrics in [25].…”

A Yau-Tian-Donaldson correspondence on a class of toric fibrations

“…The motivation for studying (2) is that many natural geometric problems in Kähler geometry correspond to (2) for suitable choices of v and w. For example, for T a maximal torus in Aut red (M ), v ≡ 1 and w ext a certain affine-linear function on t * , the (1, w ext )-cscK metrics are the extremal metrics in the sense of Calabi. Another example, which will be the one of the main interest of this paper, is the existence theory of extremal Kähler metrics on a class of toric fibrations, which can be reduced to the study of (v, w)-cscK on the toric fiber for suitable choices of v and w. Weighted Kähler metrics have been extensively studied and related to a notion of (v, w)-weighted K-stability, see for example [7,9,11,47,49].…”

“…The existence of an extremal Kähler metric in a given Kähler class is conjecturally equivalent to a certain notion of stability through an extension of the Yau-Tian-Donaldson's (YTD) conjecture, introduced [64,65] for the polarized case and [32] for a general Kähler class. This conjecture, its ramification [64,65] and extension [7,28,47,49,64,65,68] have generated tremendous efforts in Kähler geometry and have led to many interesting developments during the last decades.…”