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Solving the Kodaira-Spencer problem using harmonic analysis on torus bundles over S1
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Holt, Thomas (2022) Solving the Kodaira-Spencer problem using harmonic analysis on torus bundles over S1. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3853253
Abstract
In this thesis we will consider the spaces of ∂¯ and Bott-Chern harmonic differential forms Hp,q ∂¯ & Hp,q BC, defined on an almost complex manifold equipped with a metric compatible with the almost complex structure. In 1954, Kodaira and Spencer asked whether the Hodge numbers h p,q ∂¯ := dim H p,q ∂¯ are all invariant of the choice of metric. We will answer this question in the negative. Furthermore, in the case of compact almost complex 4- manifolds we will give a full account of the values of p and q for which both h p,q ∂¯ and h p,q BC := dim H p,q BC are or are not independent of the metric.
Specifically, we find examples of compact 4-manifolds where h 0,1 ∂¯ , h2,1 ∂¯ , h2,1 BC and h 1,2 BC all change depending on the metric, even if we restrict ourselves to the special class of almost K¨ahler metrics. We also show that the only possible values for h 1,1 ∂¯ are b− and b− + 1, while the value of h 1,1 BC is always b− + 1. Here b− denotes an invariant given by the number of d-harmonic anti-self-dual 2-forms.
In order to obtain these results, we are required to solve a system of partial differential equations. We therefore introduce a decomposition of L 2 functions on torus bundles over S 1 which allows us to rewrite this system into a family of ordinary differential equations, which we can solve by describing the Stokes phenomenon, and a family of algebraic equations, which are equivalent to the Gauss circle problem.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Harmonic Analysis, Torus (Geometry), Four-manifolds (Topology), Topological manifolds., Manifolds (Mathematics), Symplectic geometry., Geometry, Differential | ||||
Official Date: | January 2022 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Zhang, Weiyi | ||||
Format of File: | |||||
Extent: | v, 84 leaves : illustrations | ||||
Language: | eng |
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