Scalar curvature and Futaki invariant of Kähler metrics with cone singularities along a divisor

Abstract

We study the scalar curvature of Kähler 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 Kähler–Einstein Fano manifolds with the second Betti number 1, momentum-constructed constant scalar curvature Kähler metrics with cone singularities along the ∞-section exist if and only if the log Futaki invariant vanishes on the fibrewise 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.