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

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.