Scalar curvature and Futaki invariant of K\"ahler metrics with cone singularities along a divisor

TL;DR

This paper investigates scalar curvature of K"ahler metrics with cone singularities along divisors, establishing conditions for existence of constant scalar curvature metrics related to the vanishing of the log Futaki invariant, supporting a conjecture in complex geometry.

Contribution

It provides new existence criteria for conically singular K"ahler metrics with constant scalar curvature based on the log Futaki invariant, advancing the understanding of the Yau--Tian--Donaldson conjecture.

Findings

Existence of cscK metrics with cone singularities characterized by log Futaki invariant.

Scalar curvature extended as a current on the manifold for these metrics.

Partial invariance results for conically singular K"ahler metrics.

Abstract

We study the scalar curvature of K\"ahler metrics that have cone singularities along a divisor, with a particular focus on certain specific classes of such metrics that enjoy some curvature estimates. Our main result is that, on the projective completion of a pluricanonical bundle over a product of K\"ahler--Einstein Fano manifolds with the second Betti number 1, momentum-constructed constant scalar curvature K\"ahler metrics with cone singularities along the $\infty$-section exist if and only if the log Futaki invariant vanishes on the fibrewise $\mathbb{C}^*$-action, giving a supporting evidence to the log version of the Yau--Tian--Donaldson conjecture for general polarisations. We also show that, for these classes of conically singular metrics, the scalar curvature can be defined on the whole manifold as a current, so that we can compute the log Futaki invariant with respect to them. Finally, we prove some partial invariance results for them.