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

Abstract

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