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 $\infty$-section exist if and only if the log Futaki invariant vanishes on the fibrewise ${\mathbb{C}}^{\ast }$-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.

47 pages, 2 figures. To appear in Ann. Inst. Fourier

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Cited by in corpus (2)

• Construction of constant scalar curvature Kähler cone metrics
• Geodesics in the space of Kahler cone metrics, II. Uniqueness of constant scalar curvature Kahler cone metrics