Isolated Singularities of Affine Special Kähler Metrics in Two Dimensions

Abstract

We prove that there are just two types of isolated singularities of special Käh-ler metrics in real dimension two provided the associated holomorphic cubic form does not have essential singularities. We also construct examples of such metrics.

Local Models of Isolated Singularities for Affine Special Kähler Structures in Dimension Two

“…Write Ξ = Ξ 0 dz 3 , where Ξ 0 is a holomorphic function on B * 1 . By the proof of Theorem 1.1 of [Hay15] u satisfies ∆u = 16|Ξ 0 | 2 e 2u .…”

“…The main purpose of this section is to fix notations. Details can be found for instance in [Fre99,Hay15]; See also [CH17] for an elementary exposition.…”

“…Isolated singularities of affine special Kähler structures in the simplest case of two real dimensions are in the focus of this article, which is closely related to [Hay15]. However, here we focus on the properties of ∇ rather than g near an isolated singularity.…”

“…A number of special Kähler structures with isolated singularities can be found [Hay15,CH17]. We now describe a family of examples, whose significance will be clear below.…”

“…Choosing a local holomorphic coordinate z near m 0 , we can write g = w |dz| 2 and ∇ = d + ω ∇ . By the main theorem of [Hay15] there are two possibilities: w = −|z| n+1 log |z|(C + o(1)) or there is β < n + 1 such that w = |z| β (C + o(1)). In this article we prove the following result.…”