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A Special Type of Kähler Manifold
Author(s) -
Hodge W. V. D.
Publication year - 1951
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-1.1.104
Subject(s) - mathematics , type (biology) , manifold (fluid mechanics) , citation , algebra over a field , combinatorics , pure mathematics , computer science , library science , engineering , geology , mechanical engineering , paleontology
1. IN my book on harmonic integrals (4) I applied the theory of harmonic integrals to derive certain results concerning algebraic varieties. Subsequently (5) I pointed out that many of the results obtained depended only on local properties of the metric used and that some of them depended only on the fact that an algebraic variety is capable of carrying a Kahler metric, and not on the actual metric selected. Weil (7) pointed out that the results obtained in this way for algebraic varieties could be extended to apply to any Kahler manifold, and in a series of notes in the Comptes Rendus (3) Eckmann and Guggenheimer have shown how the arguments can be carried through in detail for a general Kahler manifold. The topological properties of Kahler manifolds deduced by Eckmann and Guggenheimer are similar to those obtained by Lefschetz (6) for algebraic varieties, but are weaker, since in the case of Kahler manifolds the cycles are considered relative to the field of complex numbers, whereas in the Lefschetz theory they are combined with integral coefficients. The object of the present paper is to show that if a certain restriction is imposed on the Kahler manifold, the methods of Eckmann and Guggenheimer lead to results on the integral topology of the manifold similar to those obtained by Lefschetz for algebraic varieties, though they are still incapable of taking account of torsion. The restriction imposed has other consequences, which will also be described. In order to describe the methods and to introduce the notation, it is necessary to describe in some detail the main results of Eckmann and Guggenheimer. This will be done as briefly as possible, and proofs will not be given at this stage. The opportunity will be taken to add to the results of Eckmann and Guggenheimer another result (Theorem I) of considerable interest. The proof of this result for algebraic varieties was indicated in (5), and the proof for general Kahler manifolds given below is, like the proofs of most of the theorems for Kahler manifolds which are derived by the use of harmonic integrals, similar to that for algebraic varieties.