Kähler metrics with constant weighted scalar curvature and weighted K-stability

• Published in: arXiv.org 2019-04
• Main Author: Lahdili, Abdellah
• Format: Article
• Language: English
• Subjects: Automorphisms; Blocking; Curvature; Invariants; Lie groups; Toruses; Weight
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1808.07811

We introduce a notion of a K\"ahler metric with constant weighted scalar curvature on a compact K\"ahler manifold \(X\), depending on a fixed real torus \(\mathbb{T}\) in the reduced group of automorphisms of \(X\), and two smooth (weight) functions \(\mathrm{v}>0\) and \(\mathrm{w}\), defined on the momentum image (with respect to a given K\"ahler class \(\alpha\) on \(X\)) of \(X\) in the dual Lie algebra of \(\mathbb{T}\). A number of natural problems in K\"ahler geometry, such as the existence of extremal K\"ahler metrics and conformally K\"ahler, Einstein--Maxwell metrics, or prescribing the scalar curvature on a compact toric manifold reduce to the search of K\"ahler metrics with constant weighted scalar curvature in a given K\"ahler class \(\alpha\), for special choices of the weight functions \(\mathrm{v}\) and \(\mathrm{w}\). We show that a number of known results obstructing the existence of constant scalar curvature K\"ahler (cscK) metrics can be extended to the weighted setting. In particular, we introduce a functional \(\mathcal M_{\mathrm{v}, \mathrm{w}}\) on the space of \(\mathbb{T}\)-invariant K\"ahler metrics in \(\alpha\), extending the Mabuchi energy in the cscK case, and show (following the arguments of Li and Sano--Tipler in the cscK and extremal cases) that if \(\alpha\) is Hodge, then constant weighted scalar curvature metrics in \(\alpha\) are minima of \(\mathcal M_{\mathrm{v},\mathrm{w}}\). Motivated by the recent work of Dervan--Ross and Dyrefelt in the cscK and extremal cases, we define a \((\mathrm{v},\mathrm{w})\)-weighted Futaki invariant of a \(\mathbb{T}\)-compatible smooth K\"ahler test configuration associated to \((X, \alpha, \mathbb{T})\), and show that the boundedness from below of the \((\mathrm{v},\mathrm{w})\)-weighted Mabuchi functional \(\mathcal M_{\mathrm{v}, \mathrm{w}}\) implies a suitable notion of a \((\mathrm{v},\mathrm{w})\)-weighted K-semistability.