论文标题
关于几乎kähler歧管和乔伊斯的示例的形式
On the formality of nearly Kähler manifolds and of Joyce's examples in $G_2$-holonomy
论文作者
论文摘要
这是一个突出的猜想（与Riemannian的几何形状和代数拓扑相关），所有简单地连接的特殊自治的紧凑型歧管都应是正式的空间，即，应从其理性同学代数中衍生出其合理同质类型，这是在理性同源理论中的特定属性。现在，特殊利益在于卓越的自动企业$ g_2 $和$ spin（7）$。在本文中，我们提供了一种方法，即如何确认乔伊斯著名的固体例子$ g_2 $确实是正式的空间；我们具体地为一个示例施加了此计算，该计算可能是其余乔伊斯示例的蓝图（可能也可能是固体$ spin（7）$）。在这些考虑因素之前，还有另一个结果，确定了歧管的形式承认特殊结构：我们证明了几乎kähler歧管的形式。可以找到这两个结果之间的联系，这是因为“特殊的全能”和“几乎是Kähler”自然概括了紧凑的Kähler歧管，其形式是古典的，由Deligne-Griffiths-Morgan-Morgan-Sullivan的经典定理。
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.
