Constant scalar curvature Kaehler metrics on ramified Galois coverings

TL;DR

This paper establishes conditions under which ramified Galois coverings of cscK manifolds admit Kähler-Einstein or cscK metrics, extending previous existence results through cohomological criteria.

Contribution

It provides new sufficient cohomological conditions for the existence of Kähler-Einstein and cscK metrics on ramified Galois coverings, generalizing prior results.

Findings

Derived cohomological conditions for metric existence

Extended previous Kähler-Einstein and cscK existence results

Applicable to a broader class of ramified coverings

Abstract

We give sufficient conditions for the existence of Kaehler-Einstein and constant scalar curvature Kaehler (cscK) metrics on finite ramified Galois coverings of a cscK manifold in terms of cohomological conditions on the Kaehler classes and the branching divisor. This result generalizes previous work on Kaehler-Einstein metrics by Li-Sun [Comm. Math. Phys. 2014], and extends Chen-Cheng's existence results for cscK metrics in [J. Amer. Math. Soc. 2021].