Abstract
In this chapter we define the notions of a Kuranishi structure and of a good coordinate system. We also study embedding between them, which describes a relation among those structures.
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Notes
- 1.
We might require only a weaker condition that \(\det {\mathcal E}^{\ast } \otimes \det TU\) is orientable. One of the reasons we take the current choice is that then Condition 23.9 is automatically satisfied.
- 2.
In Definition 3.1 the name, Kuranishi neighborhood, is also used for the underlying orbifold of the Kuranishi chart. Here we use it for a Kuranishi chart such that Im(ψ) contains a given subset of X. It should be easy to distinguish them from the context.
- 3.
Note that Im(ψ p) is open in X.
- 4.
Note it may happen that \(U_{\mathfrak p \mathfrak q} = \emptyset \).
- 5.
Here \(\mathfrak p <\mathfrak q\) means \(\mathfrak p \le \mathfrak q\) and \(\mathfrak p \ne \mathfrak q\).
- 6.
Note that under this condition \(\Phi ^{\prime }_{pq}\) exists by Item (1).
- 7.
- 8.
The proof of continuity is the same as Remark 3.41.
- 9.
There are various other versions proposed by other authors. We do not mention them unless they are directly related to the discussion here.
- 10.
Both by other authors and by us.
- 11.
- 12.
Actually in our recent article [FOOO21] presenting the detail of the construction of the Kuranishi structure on the moduli space of pseudo-holomorphic disks, we used an ambient set. While writing the article we found it is rather cumbersome and technically involved work to define a topology on this ambient set and prove its nice property. So we did not try to do so. See [FOOO21, Remark 7.9 (2)]. Proving Hausdorffness of the moduli space itself is much easier in comparison.
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Fukaya, K., Oh, YG., Ohta, H., Ono, K. (2020). Kuranishi Structures and Good Coordinate Systems. In: Kuranishi Structures and Virtual Fundamental Chains. Springer Monographs in Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-15-5562-6_3
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