具有平凡陈类的流形 II：与环面乘积同源的流形、余标架流形及 Baldassarri 的一个问题
Manifolds with trivial Chern classes II: Manifolds Isogenous to a Torus Product, coframed Manifolds and a question by Baldassarri
AI总结：
本文受 Baldassarri 问题启发，引入与 k-环面乘积同源的流形和部分余标架流形等新概念，刻画了二维情形下陈类平凡的正确曲面类，并指出高维情形更为复杂，同时提出若干开放问题。
摘要：
受 Mario Baldassarri 在 1956 年提出的一个一般性问题的启发，我们讨论了 Roth 引入的伪阿贝尔簇的刻画，并引入第一个新概念：与 k-环面乘积同源的流形。后者在有理上同调中最后 k 个陈类平凡，且陈数为零。我们证明在二维情形下，后一类是对 Enriques、Dantoni、Roth 和 Baldassarri 某些不正确断言的正确替代：这些曲面满足 $K_X$ nef 且 $c_2(X)=0 \in H^4(X, \mathbb{Z})$。我们在最后一节中利用 Chad Schoen 的一个构造观察到，在更高维情形下并不存在如此简单的类似图景。然后我们讨论作为 Baldassarri 问题的一类解：与射影（分别为 Kähler）流形同源的流形，其切丛或余切丛具有正秩的平凡子丛。我们看到，在 $K_X$ nef 的情形下，'部分标架'射影流形（即其切丛具有平凡子丛）的类由 Roth 的伪阿贝尔簇组成；而'部分余标架'射影流形的类尽管我们在此能够展示新的结果，但仍未被完全理解：我们提出了一些开放问题和猜想。在本文过程中，我们还处理了更一般紧复流形的情形，引入了可平行化流形上的 suspension、扭曲超椭圆流形的新概念，并描述了在 Kähler 假设下的已知结果。
英文摘要：
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.
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