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Families of Polarized Abelian Varieties and a Construction of Kahler Metrics of Negative Holomorphic Bisectional Curvature on Kodairasurfaces



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Families of Polarized Abelian Varieties and a Construction of Kahler Metrics of Negative Holomorphic Bisectional Curvature on Kodairasurfaces by Ho-Yu Tsui
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This dissertation, "Families of Polarized Abelian Varieties and a Construction of Kahler Metrics of Negative Holomorphic Bisectional Curvature on Kodairasurfaces" by Ho-yu, Tsui, 徐浩宇, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Abstract of thesis entitled FAMILIES OF POLARIZED ABELIAN VARIETIES AND A CONSTRUCTION OF KAHLER METRICS OF NEGATIVE HOLOMORPHIC BISECTIONAL CURVATURE ON KODAIRA SURFACES submitted by Tsui Ho Yu for the degree of Master of Philosophy at The University of Hong Kong August, 2006 A Kodaira surface is a family of compact complex curves with none of its fiber isomorphic to each other. Such surfaces are known to have ample cotangent bundle in the language of Algebraic Geometry. In terms of Complex Differential Geometry, thiscorrespondstothestatementthatKodairasurfacesadmitcomplex Finsler metrics of negative curvature. In 1989, I-H. Tsai proved the stronger statement that Kodaira surfaces are negative in the sense of Griffiths. In other words, theyadmitHermitianmetricsofnegativecurvatureinsomenaturalsense. In view of the special role played by K]ahler metrics in Complex Differential Geometry, it is natural to ask further whether there are any Kodaira surfaces admitting K]ahler metrics of negative holomorphic bisectional curvature. This has remained an unsolved problem for a long time.Inthisthesis, aKodairasurfaceisregardedasabranchedcycliccoveringover the product space of two non-singular Riemann surfaces. Suppose that there is no hyperelliptic curve on each fiber of the Kodaira surface, so that the second fundamental form of the embedding of each fiber of the Kodaira surface into its Jacobian variety is everywhere nonzero. By piecing canonical metrics and considering the second fundamental form, it can be shown that the holomorphic bisectional curvature of a suitable positive linear combination of these metrics is strictly negative everywhere. The first chapter consists of preliminaries on Complex Differential Geometry. An introduction to Riemannian, Hermitian and K]ahler metrics is given. The second chapter consists of a classification of Hermitian symmetric manifolds of the non-compact type. One of these spaces is a parameter space of the families of principally polarized Abelian varieties. The third chapter provides backgrounds on families of principally polarized Abelian varieties. Their differential geometric structure is discussed. Finally, examples of Kodaira surfaces and a proof of the main theorem are given in the last chapter. DOI: 10.5353/th_b3705376 Subjects: Manifolds (Mathematics)Abelian varietiesKahlerian structures
Release date NZ
January 27th, 2017
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Country of Publication
United States
colour illustrations
Open Dissertation Press
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