On the Classification of Certain Singular Hypersurfaces in CP

Hypersurfaces in complex projective spaces defined by homogeneous polynomials are important topological objects, arising naturally in algebra, geometry and topology. It was first noted by Thom, the diffeomorphism type of smooth hypersurfaces depends only on the degree of the defining polynomial; i.e., two n-dimensional smooth hypersurfaces in CPn+1 are diffeomorphic if and only if they have equal degree. For singular hypersurfaces, the situation is more complicated. A theoretical solution to the classification problem of singular hypersurfaces is [4]: let P (n, d) be the moduli space of hypersurfaces of degree d in CPn, then there is a Whitney stratification on P (n, d), such that two pairs (CPn, Vf ) and (CP n, Vg) are topologically equivalent if the hypersurfaces Vf and Vg belong to the same connected component of a stratum of this stratification. Instead of considering the pair (CPn, V ), in this paper we consider the classification of singular hypersurfaces as topological spaces. Since the nonsingular part of a hypersurface is a smooth manifold, it is natural to ask for a classification of singular hypersurfaces upto homeomorphisms, where the homeomorphisms are diffeomorphisms on the nonsingular parts. In this paper the hypersurfaces in CP4 with an isolated singularity are studied, and with some restrictions on the link of the singularity a classification in the above sense is obtained. The main result is the following: