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Solving the KodairaSpencer problem using harmonic analysis on torus bundles over S1
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Holt, Thomas (2022) Solving the KodairaSpencer 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 BottChern 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 4manifolds 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 dharmonic antiselfdual 2forms.
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)  

Subjects:  Q Science > QA Mathematics  
Library of Congress Subject Headings (LCSH):  Harmonic Analysis, Torus (Geometry), Fourmanifolds (Topology), Topological manifolds., Manifolds (Mathematics), Symplectic geometry., Geometry, Differential  
Official Date:  January 2022  
Dates: 


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Publication Status:  Unpublished  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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