Manifolds with Trivial Chern Classes II: Manifolds Isogenous to a Torus Product, Coframed Manifolds and a Question by Baldassarri

Abstract Mario Baldassarri asked in 1956 to describe the projective manifolds with last k Chern classes trivial in real cohomology. He claimed that the solutions are Roth’s pseudo-abelian varieties, while indeed the class of solutions is larger, it includes the class introduced here of manifolds isogenous to a k-torus product: these have also vanishing Chern numbers, and in dimension 2 are all the solutions with $K_{X}$ nef. We show that such a simple picture does not hold in higher dimension. Other solutions to Baldassarri’s question are the manifolds isogenous to k-framed or k-coframed manifolds, that we investigate here: the k- framed projective manifolds with $K_{X}$ nef are the pseudo-abelian varieties. We show results for the k-coframed manifolds, pose open questions and conjectures, treat also the nonprojective case.