Construction of constant scalar curvature Kähler cone metrics

Abstract

Abstract. Over a compact Kähler manifold, we provide a Fredholm alternative result for the Lichnerowicz operator associated to a Kähler metric with conic singularities along a divisor. We deduce several existence results of constant scalar curvature Kähler metrics with conic singularities: existence result under small deformations of Kähler classes, existence result over a Fano manifold, existence result over certain ruled manifolds. In this last case, we consider the projectivisation of a parabolic stable holomorphic bundle. This leads us to prove that the existing Hermitian-Einstein metric on this bundle enjoys a regularity property along the divisor on the base.

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

“…Since we can expand τ and ϕ(τ ) in the powers of |z 1 | 2β , we see from the estimates for g ϕ that the Kähler potential for ω ϕ is an element of C 4,α,β as defined e.g. in [29,36].…”

Geodesics in the Space of Kähler Cone Metrics II: Uniqueness of Constant Scalar Curvature Kähler Cone Metrics

“…The proof of this lemma is similar to the proof of the Proposition 3.1 in [44]. We could replace W 2,p,β s with C 1,α,β and use the compactness argument of C 1,α,β .…”

“…We define the spaces C 4,α,β (ω) in Section 2 in [47] and Section 2 in [44], the function in which has all their 4th order derivatives in C 0,α,β . The idea is firstly to define a local 4th order Hölder space with respect to the flat cone metric ω β near the cone points.…”

“…• With half angle restriction (5.3), the Fredholm alternative theorem above is proved in [44] in the better space C 4,α,β (ω).…”

“…The proof of the linear theory updates the method developed in our paper [44], where this theorem was proved under angle restriction 0 < β < 1 2 , but in the space C 4,α,β (ω) which has better regularities. The key steps of the proof of Theorem 5.1 can be summarised as follows:…”