Existence of weak conical Kähler–Einstein metrics along smooth hypersurfaces

Abstract

The existence of weak conical Kähler–Einstein metrics along smooth hypersurfaces with cone angle between \(0\) and \(2\pi \) is obtained by studying a family of Aubin’s (J Funct Anal 57:143–153, 1984) continuity paths and obtaining a uniform \(C^2\) estimate by a local Moser’s iteration technique. As soon as the \(C^0\) estimate is achieved, the local Moser’s iteration technique could improve the rough \(C^2\) estimate in Chen et al. (J Am Math Soc 28:183–197, 2015) to a uniform \(C^2\) estimate. Since in the cases of negative and zero Ricci curvature, the \(C^0\) estimate is unobstructed, the weak conical Kähler–Einstein metrics are obtained; while in the case of positive Ricci curvature, the \(C^0\) estimate is achieved under the assumption of the properness of the Twisted K-Energy. The method used in this paper does not depend on the bound of the holomorphic bisectional curvature of any global background conical Kähler metrics.