Abstract
In Chap. 16 we formulated various versions of corner compatibility conditions. To prove the results stated in Chap. 16 (and which also appear elsewhere in this book and will appear in the future) we need to extend the Kuranishi structure given on the boundary ∂X satisfying corner compatibility conditions to one on X.
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Notes
- 1.
Note that by construction the Kuranishi map of the outer collaring is constant in the normal direction of the collar added. So in general the outer collaring cannot be isomorphic to the original Kuranishi structure. The fact that the Kuranishi map of the outer collaring is constant in the normal direction of the collar is used in the proof of Lemma-Definition 17.14 (8) for example. However, it seems likely that we can modify the argument and can avoid the usage of this requirement.
- 2.
The map φ 12 in Definition 25.8 corresponds to the map \((\overline y,(t_1,\dots ,t_k)) \mapsto \gamma \cdot (\overline y,(t_1,\dots ,t_k))\) here. The i-th normal coordinate of \(\gamma \cdot (\overline y,(t_1,\dots ,t_k))\) is denoted by \(\widehat \varphi _{\sigma _{\gamma }^{-1}(i)}^{\gamma }(\overline y,(t_1,\dots ,t_k))\).
- 3.
Definition 25.3 (1).
- 4.
Definition 25.3 (2).
- 5.
Lemma 17.21 implies that this is an equivalence relation.
- 6.
We recall that for an orbifold chart (V, Γ, ϕ) at p we required (in Definition 23.1 (1)) that p = ϕ(o p) and p is fixed under the Γ-action.
- 7.
We remark that \(\widehat {S}_k(U) = \widehat {S}_k(U_0)\) for k ≥ 1.
- 8.
They are defined on \(\widehat {S}_k(U) = \widehat {S}_k(U_0)\).
- 9.
Since we are constructing the space U together with its orbifold structure, we need to check the cocycle condition. It is easy to check, however.
- 10.
When a τ-collared CF-perturbation \(\widehat {\mathfrak {S}}\) on \(\widehat {\mathcal {U}}\) varies in a uniform family, we may take the induced CF-perturbation \(\widehat {\mathfrak {S}'}\) to be uniform.
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Fukaya, K., Oh, YG., Ohta, H., Ono, K. (2020). Extension of a Kuranishi Structure and Its Perturbation from Boundary to Its Neighborhood. In: Kuranishi Structures and Virtual Fundamental Chains. Springer Monographs in Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-15-5562-6_17
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