Affine special Kaehler structures in real dimension two

TL;DR

This paper reviews affine special Kaehler structures in real dimension two, classifies their isolated singularities, and constructs continuous families with such singularities on the projective line.

Contribution

It provides a complete classification of isolated singularities and constructs new families of special Kaehler structures with singularities in dimension two.

Findings

All possible isolated singularities are described.

Monodromy of the flat symplectic connection near singularities is computed.

Continuous families of structures with singularities are constructed on the projective line.

Abstract

We review properties of affine special Kaehler structures focusing on singularities of such structures in the simplest case of real dimension two. We describe all possible isolated singularities and compute the monodromy of the flat symplectic connection, which is a part of a special Kaehler structure, near a singularity. Beside numerous local examples, we construct continuous families of special Kaehler structures with isolated singularities on the projective line.