ScalingStacks

Corollary 3.23 . [04IT]

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Corollary 3.23.

In view of Remark 3.21, given (B,Δ,𝒜)(B,\Delta,\mathscr{A}) as in Example 3.16, a compactification X⁡(B0,𝒜)↪(X,ω)X(B_{0},\mathscr{A})\hookrightarrow(X,\omega) as above is uniquely determined up to symplectic conjugation by a choice of 24 formal power series in two variables:

𝔮1,…,𝔮24∈ℝ⁡[[x,y]]\mathfrak{q}_{1},\ldots,\mathfrak{q}_{24}\in\mathbb{R}[\![x,y]\!]

corresponding to germs of focus-focus fibrations ℱ1,…​ℱ24\mathcal{F}_{1},\ldots\mathcal{F}_{24}. In particular, there are infinitely many Lagrangian fibrations of a symplectic K3 surface, fibering over (B,Δ,𝒜)(B,\Delta,\mathscr{A}), which are all topologically conjugate but not symplectically conjugate.

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