Constant scalar curvature Kähler metrics on ramified Galois coverings

Abstract

Abstract We give sufficient conditions for the existence of Kähler–Einstein and constant scalar curvature Kähler (cscK) metrics on finite ramified Galois coverings of a cscK manifold in terms of cohomological conditions on the Kähler classes and the branching divisor. This result generalizes previous work on Kähler–Einstein metrics by Li and Sun [C. Li and S. Sun, Conical Kähler–Einstein metrics revisited, Comm. Math. Phys. 331 2014, 3, 927–973], and extends Chen–Cheng’s existence results for cscK metrics in [X. Chen and J. Cheng, On the constant scalar curvature Kähler metrics (II)—Existence results, J. Amer. Math. Soc. 34 2021, 4, 937–1009].