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In mathematics, the Gauss–Manin connection, introduced by , is a connection on a certain vector bundle over a family of algebraic varieties. The base space is taken to be the set of parameters defining the family, and the fibers are taken to be the de Rham cohomology group of the fibers ''V''. Flat sections of the bundle are described by differential equations; the best-known of these is the Picard–Fuchs equation, which arises when the family of varieties is taken to be the family of elliptic curves. In intuitive terms, when the family is locally trivial, cohomology classes can be moved from one fiber in the family to nearby fibers, providing the 'flat section' concept in purely topological terms. The existence of the connection is to be inferred from the flat sections. ==Example== A commonly cited example is the Dwork construction of the Picard–Fuchs equation. Let : be the projective variety describing an elliptic curve. Here, is a free parameter describing the curve; it is an element of the complex projective line (the family of hypersurfaces in ''n'' − 1 dimensions of degree ''n'', defined analogously, has been intensively studied in recent years, in connection with the modularity theorem and its extensions).〔http://www.math.princeton.edu/~nmk/dworkfam64.pdf〕 Thus, the base space of the bundle is taken to be the projective line. For a fixed in the base space, consider an element of the associated de Rham cohomology group : Each such element corresponds to a period of the elliptic curve. The cohomology is two-dimensional. The Gauss–Manin connection corresponds to the second-order differential equation : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Gauss–Manin connection」の詳細全文を読む スポンサード リンク
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