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Title: Metric collapsing on calabi-yau 3-folds
Author: Li, Yang
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2019
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The aim of this thesis is to describe the collapsing Calabi-Yau (CY) metrics on a CY 3-fold with a Lefschetz K3 fibration, from both the a priori estimate perspective and the gluing perspective. Collapsing CY metric 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, mostly relying on very favourable gluing ansatz. From the nonlinear perspective, the essential realisation is that by restricting the type of singularities, and under some conjecture in pluripotential theory, then a small neighbourhood of the singular fibre has a local noncollapsing bound, which enables us to understand the pointed Gromov-Hausdor limit of the singular fibre in the scale where the fibre volume is 1. This gives strong heuristic evidence that there is a much finer scale near the nodal points in the fibration, where the scaled limit is a CY metric on C3 with maximal volume growth and singular tangent cone at infinity. This model CY metric is rigorously constructed using a noncompact version of Yau's solution to the Calabi conjecture. The model metric enables a gluing description of the collapsing CY metrics. The diffi culty 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 Sz eklyhidi.
Supervisor: Donaldson, Simon ; Haskins, Mark Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral