ScalingStacks

Theorem 6.3 [032B]

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Theorem 6.3

Let j:J→Bj:J\rightarrow B be an elliptically fibred K3 surface with a section and singular fibres all of type I1I_{1}, and let fi:Xi→Bf_{i}:X_{i}\rightarrow B be a sequence of elliptically fibred K3 surfaces with jacobian jj. Let ωi\omega_{i} correspond to a Ricci-flat Kähler metric on XiX_{i} with ωi2\omega_{i}^{2} independent of i\,i, and with ∫fi−1​(b)ωi=ϵi→0\int_{f_{i}^{-1}(b)}\omega_{i}=\epsilon_{i}\rightarrow 0 as i→∞i\rightarrow\infty. Then the sequence of Riemannian manifolds (Xi,ϵi​ωi)(X_{i},\epsilon_{i}\omega_{i}) converges in the Gromov–Hausdorff sense to BB, the metric on BB being induced from the (singular) Riemannian metric given, in local coordinates, by W0−1​d​y⊗d​y¯W_{0}^{-1}dy\otimes d\bar{y}, with W0W_{0} as defined in §4.

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