Communications in Analysis and Geometry

Volume 13 (2005)

Number 2

Singular Semi-Flat Calabi-Yau Metrics on S2

Pages: 333 – 361



John C. Loftin


This paper is motivated by recent work of Gross and Wilson [14], in which they construct degenerate limits of families of K3 surfaces equipped with Calabi-Yau metrics (Ricci-flat Kähler metrics). Upon proper rescaling, the metric limit of such a family is a two-dimensional sphere equipped with a Riemannian metric with prescribed singularities at 24 points. Away from singularities, this limit metric is an affine Kähler metric. In other words, there are natural affine flat coordinates (aj) and a local potential function with specified metric. In this case, the metric is naturally a real slice of a Calabi-Yau metric. We refer to such a metric as a semi-flat Calabi-Yau metric. Such singular semi-flat Calabi-Yau metrics on surfaces were first constructed by Greene-Shapere-Vafa-Yau [11]. We construct many examples of such metrics.

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