WE should like to acknowledge the helpful suggestions of David Arrowsmith who has read previous versions of this paper. We would also like to thank Shaun Bullett and Chris Penrose with whom we have had useful discussions about complex maps, and Nik Buric who has helped greatly in the clarification of various points. Thanks also go to Carl Murray for help in producing some of the illustrations. JHEC would like to acknowledge the support of the Science and Engineering Research Council (SERC) and the AEJMC Foundation. OP would like to acknowledge the support of the Wolfson Foundation and CONICET.