Special geometry of Calabi-Yau compactifications near a rigid limit

We discuss, in the framework of special Kahler geometry, some aspects of th e "rigid limit" of type IIB string theory compactified on a Calabi-Yau thre efold. We outline the general idea and demonstrate by direct analysis of a specific example how this limit is obtained. The decoupling of gravity and the reduction of special Kahler geometry from local to rigid is demonstrate d explicitly, without first going to a noncompact approximation of the Cala bi-Yau. In doing so, we obtain the Seiberg-Witten Riemann surfaces correspo nding to different rigid limits as degenerating branches of a higher genus Riemann surface, defined for all values of the moduli. Apart from giving a nice geometrical picture, this allows one to calculate easily some gravitat ional corrections to e.g. the Seiberg-Witten central charge formula. We mak e some connections to the 2/5 brane picture, also away from the rigid limit , though only at the formal level.