On Complex Projective Hypersurfaces which are Homology-Pn's

Summary

Abstract. We discuss hypersurfaces in P_n+1 that are homology- P_n's, i.e., they have the integral homology of P_n. The only cohomology ring- P_n's are the hyperplanes. Using singularity theoretic methods, we construct examples of homology- P_n's with isolated singularities in any dimension n ≥ 2 and for any degree d ≥ 3. In the odd-dimensional case, these are topological manifolds. Our methods yield interesting examples in singularity theory, e.g., isolated hypersurface singularity links that are topological spheres, but that are not associated to polynomials of the familiar “Pham-Brieskorn” type. Furthermore, we classify normal homology- P2 's in P3 with C^*-action up to isomorphy and homology- P₃'s in P₄ with isolated singularities up to homeomorphy, and we construct examples of homology-Pn's with non-isolated singularities.

INTRODUCTION AND STATEMENT OF RESULTS

Considering the importance of the complex projective n-space Pn = Pn(C) in algebraic geometry and topology, it is obvious that characterizing that space by algebro-geometric or topological properties always has been a matter of great interest. Therefore, it is quite natural to investigate spaces that share some of these properties. In this paper, we look for hypersurfaces in P_n+i (where n ≥ 2) with normal or even isolated singularities which are homology-P_n's, i.e., which have the same integral homology as P_n.