Manifolds with trivial Chern classes I: Hyperelliptic Manifolds and a question by Severi

TL;DR

This paper demonstrates that there exist projective hyperelliptic manifolds with trivial Chern classes that are not Abelian varieties, providing counterexamples to Severi's 1951 question and exploring their geometric properties.

Contribution

It constructs explicit examples of hyperelliptic manifolds with trivial Chern classes beyond Abelian varieties and analyzes their topological and geometric features.

Findings

Existence of non-Abelian projective hyperelliptic manifolds with trivial Chern classes.

Bagnera-de Franchis manifolds have topologically trivial tangent bundles.

Counterexamples to Severi's question about Abelian varieties with trivial Chern classes.

Abstract

We give a negative answer to a question posed by Severi in 1951, whether the Abelian Varieties are the only projective manifolds with trivial Chern classes. By Yau' s celebrated result, compact K\"ahler manifolds with trivial Chern classes must be flat, that is, they belong to the class of Hyperelliptic Manifolds (quotients $T/G$ of a complex torus $T$ by the free action of a finite group $G$). We exhibit simple examples of projective Hyperelliptic Manifolds which are not Abelian varieties and whose Chern classes are zero not only in integral cohomology, but also in the Chow ring. We prove moreover that the Bagnera-de Franchis manifolds (quotients $T/G$ as above but where the group $ G$ is cyclic) have topologically trivial tangent bundle. Our results naturally lead to the question of classifying all compact K\"ahler manifolds with topologically trivial tangent bundle, and all the counterexamples to Severi's question.