The Kodaira Classification of Surfaces

Abstract

Our goal in this and the next chapter will be to present those parts of the complex analytic and smooth classification which can be handled by classical techniques, i.e. without the use of gauge theory. Not surprisingly, these techniques are most effective when homotopy type and diffeomorphism type are essentially identical. In this chapter, we shall concentrate on the analytic aspects of surface theory, e.g. moduli, with special attention to elliptic surfaces. In the next chapter, we shall focus on the smooth topology of elliptic surfaces. Of course, such a neat division of the theory is somewhat artificial, and we shall not always strictly observe it. Thus we shall occasionally appeal here to some of the results in Chapter II. Our major concern will be to describe the “complex analytic” classification of complex surfaces, i.e. the classification up to deformation equivalence (to be precisely defined in Section 1). Two complex surfaces which are deformation equivalent are diffeomorphic, and we shall begin to investigate to what extent the converse is true.