关于近凯勒流形与 G₂-全纯中 Joyce 例子的形式性
On the formality of nearly Kähler manifolds and of Joyce's examples in $G_2$-holonomy
AI总结：
本文证明近凯勒流形是形式空间，并提出并具体演示一种计算方法，用以确认 Joyce 的 G₂ 全纯例子是形式空间，从而为特殊全纯流形形式性猜想提供支持。
摘要：
这是一个著名猜想（联系黎曼几何与代数拓扑）：所有单连通紧致特殊全纯流形都应是形式空间，即其有理同伦型应已经可由其有理上同调代数导出——这是有理同伦理论中既突出又特殊的性质。当前特别关注例外全纯 G₂ 和 Spin(7)。本文提供一种方法，用以确认著名的 Joyce 全纯 G₂ 例子确实是形式空间；我们具体对一个例子执行该计算，它可作为其余 Joyce 例子（可能也包括 Spin(7) 全纯）的蓝本。在这些考虑之前有另一个结果，识别了容许特殊结构的流形的形式性：我们证明了近凯勒流形的形式性。这两个结果之间的联系在于，“特殊全纯”与“近凯勒”都自然地推广了紧致凯勒流形，而紧致凯勒流形的形式性是 Deligne-Griffiths-Morgan-Sullivan 的经典著名定理。
英文摘要：
It is a prominent conjecture (relating Riemannian geometry and algebraic topology) that all simply-connected compact manifolds of special holonomy should be formal spaces, i.e., their rational homotopy type should be derivable from their rational cohomology algebra already -- an as prominent as particular property in rational homotopy theory. Special interest now lies on exceptional holonomy $G_2$ and $Spin(7)$. In this article we provide a method of how to confirm that the famous Joyce examples of holonomy $G_2$ indeed are formal spaces; we concretely exert this computation for one example which may serve as a blueprint for the remaining Joyce examples (potentially also of holonomy $Spin(7)$). These considerations are preceded by another result identifying the formality of manifolds admitting special structures: we prove the formality of nearly Kähler manifolds. A connection between these two results can be found in the fact that both "special holonomy" and "nearly Kähler" naturally generalize compact Kähler manifolds, whose formality is a classical and celebrated theorem by Deligne-Griffiths-Morgan-Sullivan.
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