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Complex flat manifolds and their moduli spaces

Carr, Nigel C.; (1991) Complex flat manifolds and their moduli spaces. Doctoral thesis (Ph.D), UCL (University College London). Green open access

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Abstract

Although most of the work in this thesis is algebraic, its starting point and examples come from differential topology and geometry. As essential background Chapter I includes sections which describe involuted algebras and Albert's classification of rational positively involuted algebras; representations of finite groups over fields; groups that embed in division algebras and Amitsur's classification. The differential topology of the thesis arises in the study of how one can give a flat compact Riemannian manifold a Kählerian/projective structure. In Chapter II we outline some differential geometry and the theory of Flat Riemannian manifolds, particularly holonomy, we include a description Charlap's classification. Also in this chapter we give a simple proof of a bound for the minimal dimension for a flat compact Riemannian manifold with predescribed holonomy (m(Φ) ≤ |Φ|); the proof requires Amitsur's classification. The notion of complex structures on real manifolds is introduced in Chapter III. Some work on Riemann matrices is required and given. In Chapter IV we parametrise the set of complex structures which give a (real) flat compact Riemannian manifold a Kählerian structure. A parametrisation is also given for complex structures which give a projective structure for certain manifolds with a fixed polarisation. This involves Siegel's generalised upper half plane. In Chapter V we give some examples and give the above parametrisations for certain holonomy groups and representations. Some of the working involves integral representations and cohomology of finite groups. Finally, the subject of Chapter VI is essentially independent of previous chapters in respect to the work we have done. The chapter concerns subgroups of a product of surface groups, by which we mean the fundamental group of an oriented surface of positive genus. We consider the simultaneous equivalence relations of commensurability and automorphism. In particular, we show that, in a product of two surface groups in which one factor has genus greater that one, there are infinitely many equivalence classes of normal subdirect products.

Type: Thesis (Doctoral)
Qualification: Ph.D
Title: Complex flat manifolds and their moduli spaces
Open access status: An open access version is available from UCL Discovery
Language: English
Additional information: Thesis digitised by ProQuest.
URI: https://discovery.ucl.ac.uk/id/eprint/10121697
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