Abstract

We use the gluing method to give a refined description of the collapsing Calabi–Yau metrics on Calabi–Yau 3-folds admitting a Lefschetz K3 fibration.

Highlights

  • Introduction and BackgroundLet X be a compact Calabi–Yau 3-fold with a Lefschetz K3 fibration π : X →Y = P1

  • Given a reference Kahler metric ωX on X and ωY on Y, we aim to describe the collapsing family of Calabi–Yau metrics ωt representing the Kahler class

  • We impose the volume normalisation Xy ωX2 = 1 where Xy is any fibre of π, and Y ωY = 1

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Summary

Introduction and Background

As we approach the nodal points of the fibration, at a third length scale of order ∼ t1/6 (the ‘quantisation scale’), much smaller than the diameter scale of the fibres ∼ 1, one observes that the semi-Ricci-flat description must break down [Li17] This motivates the construction of a model CY metric ωC3 on C3, whose asymptotic behaviour at infinity is designed to match up approximately with the semi-Ricci-flat metric [Li17, CR17, Sze17]. It is worth pointing out that following the recent works [CR17, Sze17], many other new examples of complete CY metrics on Cn are known, which are strong candidates for modelling collapsing fibrations with higher dimensional fibres Such examples are likely to provide a vast generalisation of the main result of the present paper. All constants are uniform for sufficiently small t unless stated otherwise

Construction of Metric Ansatz
Λ1 Λ22
Inverting the Laplacian
Its key property is that dη2
Findings
Collapsed Calabi–Yau Metric
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