The paper under review is motivated by the problem of uniqueness of constant scalar curvature Kähler (cscK for short) metrics with conical singularities along a divisor on a compact complex manifold. Previously, uniqueness has been proved for conical Kähler-Einstein metrics.
In this paper, the authors introduce a Hölder-type space \(\mathcal{C}^{4,\alpha,\beta}\) on a Kähler manifold \(X\) with a smooth divisor \(D\), which is analog to the spaces \(\mathcal{C}^{\alpha,\beta}\), \(\mathcal{C}^{2,\alpha,\beta}\) introduced by S. K. Donaldson [in: Essays in mathematics and its applications. In honor of Stephen Smale’s 80th birthday. Berlin: Springer. 49–79 (2012; Zbl 1326.32039)] on a Kähler manifold \(X\) with a smooth divisor \(D\) in relation to the study of the regularity of the Laplace operator. Then they prove that a \(\mathcal{C}^{4,\alpha,\beta}\) Kähler potential \(u\) on a cscK manifold with cone angle \(2\pi\beta\leq\pi\) is in \(\mathcal{C}^{2,\alpha,\beta}\). Under the same conditions, they also show that an analog of Matsushima-Lichnerowics theorem holds: the Lie algebra of holomorphic vector fields of \(X\) tangent to \(D\) is reductive.