K\"ahler metrics with constant weighted scalar curvature and weighted K-stability

TL;DR

This paper introduces a new concept of K"ahler metrics with constant weighted scalar curvature, extending existing theories and stability notions, and explores their existence and obstructions on compact K"ahler manifolds with applications to various geometric problems.

Contribution

It defines weighted scalar curvature and stability notions, extending classical concepts to a broader weighted setting in K"ahler geometry, and links these to existence results.

Findings

Weighted scalar curvature generalizes classical scalar curvature.

Weighted Mabuchi functional's boundedness implies weighted K-semistability.

Existence of weighted cscK metrics relates to stability conditions.

Abstract

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.