On singular variants of the Singer-Hopf Conjecture

TL;DR

This paper introduces singular variants of the Singer-Hopf conjecture based on various Euler characteristics and proves these conjectures under specific conditions related to the nefness of the cotangent bundle of the ambient variety.

Contribution

It formulates new singular variants of the Singer-Hopf conjecture and proves them under conditions involving the nefness of the cotangent bundle.

Findings

Proposes singular variants of the Singer-Hopf conjecture.

Proves the conjecture when the cotangent bundle is nef.

Extends results to varieties admitting finite morphisms to manifolds with nef cotangent bundles.

Abstract

We propose singular variants of the Singer-Hopf conjecture, formulated in terms of the Euler-Mather characteristic, intersection homology Euler characteristic and, resp., virtual Euler characteristic of a closed irreducible subvariety of an aspherical complex projective manifold. We prove the conjecture under the assumption that the cotangent bundle of the ambient variety is numerically effective (nef), or, more generally, when the ambient manifold admits a finite morphism to a complex projective manifold with a nef cotangent bundle.