Hashimoto, Y; (2015) Some results on stability and canonical metrics in Kähler geometry. Doctoral thesis , UCL (University College London).

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Abstract

In this thesis, we prove various results on canonical metrics in Kähler geometry, such as extremal metrics or constant scalar curvature Kähler (cscK) metrics, and discuss connections to the notions of algebro-geometric stability of the underlying manifold. After reviewing the background materials in Chapter 1, we discuss in Chapter 2 the extension of Donaldson’s quantisation to the case where the automorphism group is no longer discrete. This is achieved by considering a new equation which states that the (1,0)-part of the gradient of the Bergman function is a holomorphic vector field. The main result of this thesis is the asymptotic existence of solutions to this equation, assuming the existence of extremal metrics. We also prove that the sequence of these solutions approximates the extremal metric, and that the solvability of the equation implies that a polarised Kähler manifold admitting an extremal metric is asymptotically weakly Chow polystable relative to any maximal torus in the automorphism group; this stability result was originally proved by Mabuchi using a different method. In Chapter 3 we discuss Kähler metrics with cone singularities along a divisor. We provide the first supporting evidence for the log Donaldson–Tian–Yau conjecture for general polarisations, and study various properties of the log Donaldson–Futaki invariant computed with respect to conically singular metrics. In Chapter 4 we discuss canonical metrics on the blow-up of manifolds with canonical metrics. This problem is well-understood when we blow up points, but few examples are known when we blow up higher dimensional submanifolds. We prove that the projective spaces blown up along a line cannot admit cscK metrics in any polarisations, but admit an extremal metric in each Kähler class that is close to the pullback of the Fubini–Study class, with an explicit formula in action-angle coordinates.

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