Manifolds with trivial Chern classes II: Manifolds Isogenous to a Torus Product, coframed Manifolds and a question by Baldassarri

TL;DR

This paper explores manifolds with trivial Chern classes, introduces new classes like manifolds isogenous to a torus product, and investigates their properties, especially in relation to Baldassarri's 1956 question and complex geometry classifications.

Contribution

It introduces the concept of manifolds isogenous to a k-torus product and analyzes their properties, extending the understanding of Chern class triviality in complex manifolds.

Findings

In dimension 2, such manifolds are characterized as surfaces with nef canonical bundle and zero second Chern class.

Higher-dimensional analogs do not follow the same pattern, as shown by Schoen's construction.

Partially framed and co-framed projective manifolds relate to pseudo-abelian varieties and are partially understood.

Abstract

Motivated by a general question addressed by Mario Baldassarri in 1956, we discuss characterizations of the Pseudo-Abelian Varieties introduced by Roth, and we introduce a first new notion, of Manifolds Isogenous to a k-Torus Product: the latter have the last k Chern classes trivial in rational cohomology and vanishing Chern numbers. We show that in dimension 2 the latter class is the correct substitute for some incorrect assertions by Enriques, Dantoni, Roth and Baldassarri: these are the surfaces with $K_X$ nef and $c_2(X)=0 \in H^4(X, \mathbb{Z})$. We observe in the last section, using a construction by Chad Schoen, that such a simple similar picture does not hold in higher dimension. We discuss then, as a class of solutions to Baldassarri's question, manifolds isogenous to projective (respectively: K\"ahler) manifolds whose tangent bundle or whose cotangent bundle has a trivial subbundle of positive rank. We see that the class of `partially framed' projective manifolds (that is, whose tangent bundle has a trivial subbundle) consists, in the case where $K_X$ is nef, of the Pseudo-Abelian varieties of Roth; while the class of `partially co-framed' projective manifolds is not yet fully understood in spite of the new results that we are able to show here: and we formulate some open questions and conjectures. In the course of the paper we address also the case of more general compact complex Manifolds, introducing the new notions of suspensions over parallelizable Manifolds, of twisted hyperelliptic Manifolds, and we describe the known results under the K\"ahler assumption.