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Period Matrix
In mathematics, in the field of algebraic geometry, the period mapping relates families of Kähler manifolds to families of Hodge structures. Ehresmann's theorem Let be a holomorphic submersive morphism. For a point ''b'' of ''B'', we denote the fiber of ''f'' over ''b'' by ''X''''b''. Fix a point 0 in ''B''. Ehresmann's theorem guarantees that there is a small open neighborhood ''U'' around 0 in which ''f'' becomes a fiber bundle. That is, is diffeomorphic to . In particular, the composite map :X_b \hookrightarrow f^(U) \cong X_0 \times U \twoheadrightarrow X_0 is a diffeomorphism. This diffeomorphism is not unique because it depends on the choice of trivialization. The trivialization is constructed from smooth paths in ''U'', and it can be shown that the homotopy class of the diffeomorphism depends only on the choice of a homotopy class of paths from ''b'' to 0. In particular, if ''U'' is contractible, there is a well-defined diffeomorphism up to homotopy. The dif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |