Singular Semi-Flat Calabi-Yau Metrics on S^2

TL;DR

This paper constructs new semi-flat Calabi-Yau metrics with singularities on the 2-sphere, expanding understanding of affine structures and mirror symmetry in complex geometry.

Contribution

It introduces methods to generate semi-flat Calabi-Yau metrics with multiple singularities on S^2, extending previous constructions and analyzing their local affine structures.

Findings

Constructed metrics with 6 or more singularities on S^2

Computed local affine structures near singularities

Analyzed mirror symmetry actions on these surfaces

Abstract

On an affine flat manifold with coordinates x^j and convex local potential function f, we call the affine Kahler metric f_{ij} dx^i dx^j semi-flat Calabi-Yau if it satisfies det f_{ij} = 1. Recently Gross-Wilson have constructed many such metrics on S^2 minus 24 singularities, as degenerate limits of Calabi-Yau metrics on elliptic K3 surfaces. We construct many more such metrics on S^2, singular at any 6 or more points, and compute the local affine structure near the singularities. The techniques involve affine differential geometry and solving a semilinear PDE on S^2 minus singularities. We also compute the action mirror symmetry should have on the resulting surfaces.