# Projective embeddings of compact Kähler manifolds

Abstract (Summary)

(Uncorrected OCR)
Abstract of thesis entitled
PROJECTIVE EMBEDDINGS OF COMPACT KAHLER
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 Kahler manifold M which admits a positive line bundle L, there exists a ko such that for any k > ko the mapping iLk : M ? PN defined by iL k(x) = [sq(x), ???, sn(x)] is a well-defined embedding, where {so, ???, sN} is a basis of the vector space of global sections of the holomorphic line bundle Lk over M. The theorem is known as the Kodaira Embedding Theorem. The existence of global holomorphic sections of Lk is based on Kodaira? 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? solution to the Levi Problem(1958), since the unit disk bundle U of the dual of a positive line bundle L is strongly pseudoconvex and the zero section in U 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 Lk 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? solution of the Levi problem was also gone through and his result was adapted to prove the embedding for certain k.
By comparing the methods of proving projective embedding based on different areas of the theory of complex manifolds, crucial links between important notions can be observed,
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.
Bibliographical Information:

Advisor:

School:The University of Hong Kong

School Location:China - Hong Kong SAR

Source Type:Master's Thesis

Keywords:embeddings mathematics kählerian manifolds

ISBN:

Date of Publication:01/01/2005