2021/08/30 by Fu, Xin, Hein, Hans-Joachim, Jiang, Xumin · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2108.13390
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 ωh = -(n+1)i∂∂log(-log h) defines a complete Kähler-Einstein metric on V ∖ D with \rm Ric(ωh) = -ωh. In fact, ωh is complex hyperbolic, i.e., the holomorphic sectional curvature of ωh is constant, and ωh 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 \rm Ric(ω) = -ω on U ∖ D, then there exist a Hermitian metric h as above and a δ∈ ℝ+ such that ω- ωh = O(e^-δ√-log h) to all orders with respect to ωh as h → 0. This rate is doubly exponential in the distance from a fixed point, and is sharp.