Michelacakis, Nickolas John; (1995) On the Picard group of compact complex projective flat manifolds. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

We consider the problem of describing the projective imbeddings of a compact, complex, projective, flat manifold M .L.S. Charlap has studied the algebraic classification of flat manifolds and recently F.E.A. Johnson using Albert's classification of rational positively involuted, finite dimensional algebras, succeeded in giving a necessary and sufficient condition for M to admit at least one complex, algebraic structure. This was done purely and solely in terms of the rational holonomy representation of M. We carry this further and using extension theory we give a description of Pic(M) by proving a generalized equivariant version of the well known Appell-Humbert theorem. We finally classify the projective flat manifolds whose holonomy group is either cyclic Cp or dihedral D2p , where p is a prime. We use our results to provide an estimate for the size of the set of all positive line bundles and construct interesting imbeddings directly in terms of the integral holonomy representation of M.

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