Projective Embeddings of Compact Kahler Manifolds

This dissertation, "Projective Embeddings of Compact Kahler Manifolds" by Wai-hung, Lam, 林偉雄, 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 PROJECTIVEEMBEDDINGSOFCOMPACTKAHLER MANIFOLDS submitted by LAM WAI HUNG for the degree of Master of Philosophy at The University of Hong Kong in August 2004 For an arbitary compact complex manifold, the existence of any projective embedding is not guaranteed. Kodaira proved that for any compact K]ahler manifoldM which admits a positive line bundle L, there exists a k such that for any k >k the mapping i k: M → 0 0 L P definedbyi k(x) = [s (x), ---, s (x)]isawell-definedembedding, where{s, ---, s } L 0 N 0 N is a basis of the vector space of global sections of the holomorphic line bundle L over M. The theorem is known as the Kodaira Embedding Theorem. The existence of global holomorphic sections of L is based on Kodaira's vanishing theorem. On the other hand, properties of strongly pseudoconvex manifolds can be used to prove the existence of projective embeddings of certain compact complex manifolds. This resulted from H. Grauert's solution to the Levi Problem(1958), since the unit disk bundle U ofthedualofapositivelinebundleLisstronglypseudoconvexandthezerosectioninU can be blown down to give a Stein space. Considering Taylor coefficients of holomorphic functions on U, it was proven in this thesis that there are enough global holomorphic sections of L to define projective embeddings of M for certain k. The proof was self- contained and given in full details. It was derived by an elementary method working directly on the unit disk bundle without resorting to the use of Stein spaces obtained by blowing down the zero section of L . In this thesis study, the foundational materials on complex manifolds were given in order to prepare for the proof of the original version of the Kodaira Embedding Theorem. Grauert's solution of the Levi problem was also gone through and his result was adapted to prove the embedding for certain k. Bycomparingthemethodsofprovingprojectiveembeddingbasedondifferentareasof thetheoryofcomplexmanifolds, cruciallinksbetweenimportantnotionscanbeobserved, e.g. between positivity of line bundles (on compact complex manifolds) and strongly pseudoconvex domains. There is an interesting contrast between the proofs. The first proof is based on the cohomologies derived from differential forms. The second proof is based on the Cech cohomologies. The two approaches are conceptually linked to each other since Dolbeault cohomology groups correspond to Cech cohomology groups by the Dolbeault Isomorphism Theorem. DOI: 10.5353/th_b3131362 Subjects: Embeddings (Mathematics) Kahlerian manifolds