Asymptotics of Kähler-Einstein metrics on complex hyperbolic cusps

Abstract

Let L be a negative holomorphic line bundle over an (n − 1)-dimensional complex torus D. Let h be a Hermitian metric on L such that the curvature form of the dual Hermitian metric defines a flat Kähler metric on D. Then h is unique up to scaling, and, for some closed tubular neighborhood V of the zero section D ⊂ L, the form ω h = −(n + 1)i∂∂ log(−log h) defines a complete Kähler-Einstein metric on V \ D with Ric(ω h ) = −ω h . In fact, ω h is complex hyperbolic, i.e., the holomorphic sectional curvature of ω h is constant, and ω h has the usual doubly-warped cusp structure familiar from complex hyperbolic geometry. In this paper, we prove that if U is another closed tubular neighborhood of the zero section and if ω is a complete Kähler-Einstein metric with Ric(ω) = −ω on U \ D, then there exist a Hermitian metric h as above and a δ ∈ R + such that ω − ω h = O(e −δ √ −log h ) to all orders with respect to ω h as h → 0. This rate is doubly exponential in the distance from a fixed point, and is sharp.

Higher regularity for singular Kähler-Einstein metrics

“…The main progress in this direction is due to Hein-Sun [24], who showed that near a large class of smoothable isolated singularities that are locally isomorphic to a Calabi-Yau cone, the singular Calabi-Yau metric must be asymptotic in a strong sense to the Calabi-Yau cone metric. Recently an analogous result was shown by Datar-Fu-Song [15] in the case of isolated log canonical singularities using the bounded geometry method, and precise asymptotics were obtained shortly after by Fu-Hein-Jiang [20]. In more general settings the best results so far give some control of the Kähler potential, such as the work of Guedj-Guenancia-Zeriahi [22] showing continuity.…”