ScalingStacks

Theorem 4.6 . [04J2]

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Theorem 4.6.

Let ℱ=(X,ω,f,B)\mathcal{F}=(X,\omega,f,B) be a generic-singular fibration. Assume that Σ=Crit⁡(f)\Sigma=\Crit(f) is non-degenerate. Then there is a T2T^{2} invariant neighborhood U⊆XU\subseteq X of Σ\Sigma and a commutative diagram

U→ΨD4×D1×S1f|U↓↓FB→ψD2×D1\begin{CD}U@>{\Psi}>{}>D^{4}\times D^{1}\times S^{1}\\ @V{f|_{U}}V{}V@V{}V{F}V\\ B@>{\psi}>{}>D^{2}\times D^{1}\end{CD} (16)

where coordinates (x,y)(x,y) on D4D^{4} and (r,θ)(r,\theta) on D1×S1D^{1}\times S^{1} define standard symplectic coordinates, the map Ψ\Psi is a symplectomorphism, ψ\psi is a diffeomorphism sending Δ\Delta to {0}×D1\{0\}\times D^{1} and FF is given by (14). Furthermore Ψ\Psi can be taken to be Tn−1T^{n-1} equivariant.

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