Black holes, elementary strings and holomorphic anomaly

In a previous paper we had proposed a specfic route to relating the entropy of two charge black holes to the degeneracy of elementary string states in N= 4 supersymmetric heterotic string theory in four dimensions. For toroidal compactification this proposal works correctly to all orders in a power series expansion in inverse charges provided we take into account the corrections to the black hole entropy formula due to holomorphic anomaly. In this paper we demonstrate that similar agreement holds also for other N= 4 supersymmetric heterotic string compactifications.