Families of singular algebraic varieties that are rationally elliptic spaces

TL;DR

This paper studies families of hypersurfaces with isolated singularities in projective space that have finite combined ranks of rational homotopy and homology groups, revealing infinitely many homotopy types with nef canonical classes.

Contribution

It introduces a new class of hypersurfaces with specific topological properties and shows the non-existence of certain smooth rationally elliptic 3-folds.

Findings

Infinitely many homotopy types with nef canonical or anti-canonical class.

Finite sum of ranks of rational homotopy and homology groups for these families.

Non-existence of a smooth rationally elliptic 3-fold family with these properties.

Abstract

We discuss families of hypersurfaces with isolated singularities in projective space with the property that the sum of the ranks of the rational homotopy and the homology groups is finite. They represent infinitely many distinct homotopy types with all hypersurfaces having a nef canonical or anti-canonical class. In the appendix we show that such an infinite family of smooth rationally elliptic 3-folds does not exist.