The rigid limit in special Kähler geometry - from -fibrations to special Riemann surfaces: a detailed case study

Abstract

The limiting procedure of special Kähler manifolds to their rigid limit is studied for moduli spaces of Calabi-Yau manifolds in the neighbourhood of certain singularities. In two examples we consider all the periods in and around the rigid limit, identifying the non-trivial ones in the limit as periods of a meromorphic form on the relevant Riemann surfaces. We show how the Kähler potential of the special Kähler manifold reduces to that of a rigid special Kähler manifold. We make extensive use of the structure of these Calabi-Yau manifolds as K3 fibrations, which is useful to obtain the periods even before the K3 degenerates to an ALE manifold in the limit. We study various methods to calculate the periods and their properties. The development of these methods is an important step to obtaining exact results from supergravity on Calabi-Yau manifolds.