Collapsing Calabi-Yau metrics on Lefschetz K3 fibred 3-folds

Collapsing Calabi-Yau metrics on Lefschetz K3 fibred 3-folds

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Yang Li, IAS
Fine Hall 314

I will discuss the problem of describing the collapsing CY metrics on a CY 3-fold with a Lefschetz K3 fibration. Collapsing CY metrics is a well studied subject, but most of the previous works concentrate on the behaviour away from the singular fibres, and the full description of the metric was only available in a very small number of cases.

From the nonlinear perspective, the essential realisation is that by restricting the type of singularities, then a small neighbourhood of the singular fibre has a local noncollapsing bound, which enables us to understand the pointed Gromov-Hausdorff limit of the singular fibre in the scale where the fibre volume is 1.

From the gluing perspective, the main geometric insight is that there should be a much finer scale near the nodal points in the fibration, where the scaled limit is a CY metric on C^3 with maximal volume growth and singular tangent cone at infinity. This model metric was previously constructed by the author in a separate work. The difficulty of the gluing lies in the coarse nature of the gluing ansatz, and the fact that the metric has many types of characteristic behaviours at different scales. We overcome this by developing a sharp linear theory, using some earlier ideas of Gabor Szeklyhidi.