ScalingStacks

Theorem 3.3 [03L0]

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Theorem 3.3

Let XtX_{t} be a generic, nonsingular quintic in ℂ​ℙ4\mathbb{CP}^{4} near the large complex structure limit. Then Ruan’s construction yields a Lagrangian fibration f:Xt→𝒮3f:X_{t}\rightarrow{\mathcal{S}}^{3} with the following properties:

  • (i)

    ff is a piecewise smooth map.

  • (ii)

    The set of singular points in XtX_{t} of singular fibres of ff is a holomorphic curve in XtX_{t}.

  • (iii)

    The set Δ={b∈B:f−1(b)\Delta=\{b\in B:f^{-1}(b) is singular}\} is a 22-manifold with boundary in 𝒮3{\mathcal{S}}^{3}. It splits naturally into a disjoint union Δ=Δ0∪Δ1∪Δ2\Delta=\Delta_{0}\cup\Delta_{1}\cup\Delta_{2}, where Δ2\Delta_{2} is the 22-dimensional interior of Δ\Delta, and Δ1\Delta_{1} is a finite set of open intervals on the boundary of Δ\Delta, and Δ0\Delta_{0} is a finite set.

  • (iv)

    If b∈𝒮3∖Δb\in{\mathcal{S}}^{3}\setminus\Delta then f−1​(b)f^{-1}(b) is diffeomorphic to T3T^{3}.

  • (v)

    If b∈Δ2b\in\Delta_{2} then f−1​(b)f^{-1}(b) is a T3T^{3} with two isotopic circles collapsed to two singular points.

  • (vi)

    If b∈Δ1b\in\Delta_{1} then f−1​(b)f^{-1}(b) is a T3T^{3} with one circle collapsed to one singular point.

  • (vii)

    If b∈Δ0b\in\Delta_{0} then f−1​(b)f^{-1}(b) is a T3T^{3} with one T2T^{2} collapsed to one singular point.

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