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

TL;DR

This work broadens the landscape for understanding complex manifolds with vanishing top Chern data by introducing Manifolds Isogenous to a k-Torus Product (MITP) and exploring partially framed/coframed structures. It shows that in dimension two, manifolds with nef canonical class and vanishing higher Chern data align with strong isogenies to torus products (and thus with Roth’s Pseudo-Abelian varieties), while higher dimensions admit counterexamples, notably Schoen’s Calabi–Yau threefolds, that resist such simplifications. The paper develops a framework around suspensions over parallelizable manifolds, twisted hyperelliptic manifolds, Seifert fibrations, and the notions of hanging on tangent/cotangent trivial subbundles (k-framed and k-coframed), to construct broader solution classes and to pose open questions about when these imply pseudo-Abelian behavior. It also integrates historical discussion and investigates the non-Kähler setting, showing how isogeny concepts extend beyond the Kähler category and highlighting important unresolved questions regarding coframed manifolds and Abundance-type assumptions. Collectively, the results delineate when vanishing Chern data enforce a torus-product-like structure and when counterexamples reveal deeper geometric phenomena requiring new invariants and methods.

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: Kähler) 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 Kähler assumption.

Theorems & Definitions (57)

• Remark 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Conjecture 2.6
• Theorem 3.1
• Theorem 3.2
• proof
• Proposition 3.3