On complete constant scalar curvature Kähler metrics with Poincaré–Mok–Yau asymptotic property

Abstract

Abstract. Let X be a compact Kähler manifold and S a subvariety of X with higher co-dimension. The aim is to study complete constant scalar curvature Kähler metrics on non-compact Kähler manifold X − S with Poincaré-MokYau asymptotic property (see Definition 1.1). In this paper, the methods of Calabi's ansatz and the moment construction are used to provide some special examples of such metrics.

The regular quantizations of certain holomorphic bundles

“…, where c > 0 and λ = −1. According to Lemma 4.1 of [19] and the explicit expression of function F , ω can be extended across M − E.…”

Globally conformally Kähler Einstein metrics on certain holomorphic bundles

“…Proof of Theorem 1.1. From Lemma 4.1 of [16], the metric g F can be extended across M = P(E ⊕ 1) − E if and only if ϕ(x 1 ) = 0 and ϕ ′ (x 1 ) = −1.…”

Nonexistence for complete Kähler–Einstein metrics on some noncompact manifolds

“…In the Ricci-flat case, this was answered in [22] based on a theorem proved in [19] for volume of complete noncompact Riemannian manifolds. In order to handle this question, the authors (in [5,4]) introduced the concept of complete metrics with Poincaré-Mok-Yau (PMY) asymptotic property, and constructed many constant scalar curvature Kähler metrics with PMY asymptotic property on some special types of noncompact Kähler manifolds. Since these PMY type metrics are not Kähler-Einstein, naturally one can ask the following question Problem 1.2.…”

“…Several years ago, the second author [5] proposed the following questions Problem 1.1. Let M be a compact Kähler manifold and N be a subvariety with codimension bigger than or equal to 2, how to find a complete canonical metric on the noncompact Kähler manifold M − N ?…”