Kahler-Einstein metric near an isolated log canonical singularity

Abstract

We construct Kähler-Einstein metrics with negative scalar curvature near an isolated log canonical (non-log terminal) singularity. Such metrics are complete near the singularity if the underlying space has complex dimension 2 or if the singularity is smoothable. In complex dimension 2, we show that any complete Kähler-Einstein metric of negative scalar curvature near an isolated log canonical (non-log terminal) singularity is smoothly asymptotically close to one of the model metrics constructed by Kobayashi and Nakamura arising from hyperbolic geometry.Remark 3.1. The above theorem is true as long as ω KE and ω ′ KE are complete. No other metric properties of ω KE , ω ′ KE are required. We prove some corollaries of Theorem 1.3.Corollary 3.1. Suppose we are in the setting of theorem 1.3 i.e, U admits a complete Kähler-Einstein metric ω KE with negative scalar curvature and V ol ω KE (U ) < ∞, then for any other complete Kähler-Einstein metric ω ′ KE with negative scalar curvature,Another simple application is a quick proof of uniqueness of complete Kähler-Einstein metric on a complex manifold without boundary.

Asymptotics of Kähler-Einstein metrics on complex hyperbolic cusps

“…In this setting, the model metric ω h is still Kähler-Einstein, but unless D is a finite quotient of a complex torus it is not complex hyperbolic, and indeed its holomorphic sectional curvature is unbounded above and below. Unboundedness of the curvature would start causing problems in Section 2.3 (results of Datar-Fu-Song [6]) but also at various points later in the paper. Except for generalizing the Monge-Ampère C 2 estimate of [6,13], these problems should be solvable along the lines of [10], but the C 2 estimate remains a fundamental obstacle.…”

“…Unboundedness of the curvature would start causing problems in Section 2.3 (results of Datar-Fu-Song [6]) but also at various points later in the paper. Except for generalizing the Monge-Ampère C 2 estimate of [6,13], these problems should be solvable along the lines of [10], but the C 2 estimate remains a fundamental obstacle. Thus, overall, in this paper we chose to restrict ourselves to the case where D is a torus.…”

“…Write Y = ∂V . Datar-Fu-Song [6] have proved that for any smooth function β : Y → R there exists a unique complete Kähler-Einstein metric ω = ω h + i∂∂u on V such that ω n = e u ω n h on V , hence in particular Ric(ω) = −ω on V , and u = β on Y . Moreover, u and all of its covariant derivatives with respect to ω h decay like O((−log x) −1 ), i.e., linearly in terms of the distance function ρ.…”

“…Let U be a closed tubular neighborhood of the zero section D ⊂ L. Let ω be a complete Kähler-Einstein metric on U \ D such that Ric(ω) = −ω. Let u = log(ω n /ω n h ), so that ω = ω h + i∂∂u and, by [6], u = O((−log x) −1 ) as x → 0 + to all orders with respect to ω h . Then there exists a constant c ∈ R such that for all k ∈ N 0 ,…”

“…Organization of the paper. In Section 2 we carefully review the geometry of the model metrics ω h , compute the linearized operator in convenient coordinates, and state the necessary results from [6]. In Section 3, after some preliminary estimates on the solution u, we expand u according to powers of x, which determines the tangent cone.…”