Kuranishi空间的一个新定义
A new definition of Kuranishi space
摘要：
Kuranishi空间是在辛几何中的Shakaya，Oh，Ohta和Ono的工作中引入的（参见arXiv：1106.4882），作为$J$ -全纯曲线的模空间的几何结构。Kuranishi空间的另一种形式是Hessel，Wysocki和Zehnder的“polyfolds”（参见arXiv：1407.3185）。找到一个令人满意的Kuranishi空间的定义一直是最近争论的主题（参见arXiv：1208.1340，arXiv：1209.4410，arXiv：1510.06849）。我们提出了Kuranishi空间的三个新定义：一个简单的“流形”版本，“$μ$ -Kuranishi空间”，形成一个普通的范畴$\boldsymbolμ\bf Kur$ ;一个更复杂的“流形”版本，“m-Kuranishi空间”，形成一个弱2-范畴$\bf mKur$ ;和一个“orbifold”版本，“Kuranishi空间”，形成一个弱2-范畴$\bf Kur$。这些是由等价的范畴$\boldsymbolμ{\bf Kur}\simeq{\rm Ho}({\bf mKur})$，其中${\rm Ho}({\bf mKur})$是同伦范畴的$\bf mKur$，并通过一个完整的和忠实的嵌入${\bf mKur}\hookrightarrow\bf Kur$。我们还定义了（$μ$ -，m-）Kuranishi空间的边界，并与角。我们希望我们的定义将被接受为最终定义，取代以前的定义。任何一个Ekaya-Oh-Ohta-Ono Kuranishi空间$\bf X$都可以唯一地成为一个紧Kuranishi空间$\bf X'$，直到等价于$\bf Kur$（即直到同构于${\rm Ho}({\bf Kur})$）。同样适用于拓扑空间与aya-Oh-Ohta-Ono '良好的坐标系'，并为麦克达夫和Wehrheim的'Kuranishi地图集'在arXiv：1508.01556。一个在Hyndle，Wysocki和Zehnder意义下具有多重Fredholm结构的紧拓扑空间$\bf X$可以唯一地成为一个Kuranishi空间$\bf X$，直到等价于$\bf Kur$ .这本书在arXiv：1510.07444中进行了调查。
英文摘要：
'Kuranishi spaces' were introduced in the work of Fukaya, Oh, Ohta and Ono in symplectic geometry (see e.g. arXiv:1106.4882), as the geometric structure on moduli spaces of $J$-holomorphic curves. An alternative to Kuranishi spaces is the 'polyfolds' of Hofer, Wysocki and Zehnder (see e.g. arXiv:1407.3185). Finding a satisfactory definition of Kuranishi space has been the subject of recent debate (see e.g. arXiv:1208.1340, arXiv:1209.4410, arXiv:1510.06849). We propose three new definitions of Kuranishi space: a simple 'manifold' version, '$μ$-Kuranishi spaces', which form an ordinary category $\boldsymbolμ\bf Kur$; a more complicated 'manifold' version, 'm-Kuranishi spaces', which form a weak 2-category $\bf mKur$; and an 'orbifold' version, 'Kuranishi spaces', which form a weak 2-category $\bf Kur$. These are related by an equivalence of categories $\boldsymbolμ{\bf Kur}\simeq{\rm Ho}({\bf mKur})$, where ${\rm Ho}({\bf mKur})$ is the homotopy category of $\bf mKur$, and by a full and faithful embedding ${\bf mKur}\hookrightarrow\bf Kur$. We also define ($μ$-, m-)Kuranishi spaces with boundary, and with corners. We hope our definitions will become accepted as final, replacing previous definitions. Any Fukaya-Oh-Ohta-Ono Kuranishi space $\bf X$ can be made into a compact Kuranishi space $\bf X'$ uniquely up to equivalence in $\bf Kur$ (that is, up to isomorphism in ${\rm Ho}({\bf Kur})$). The same holds for topological spaces with Fukaya-Oh-Ohta-Ono 'good coordinate systems', and for McDuff and Wehrheim's 'Kuranishi atlases' in arXiv:1508.01556. A compact topological space $\bf X$ with a 'polyfold Fredholm structure' in the sense of Hofer, Wysocki and Zehnder can be made into a Kuranishi space $\bf X$ uniquely up to equivalence in $\bf Kur$. This book is surveyed in arXiv:1510.07444.
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