Gauss–Manin connection

In mathematics, the Gauss–Manin connection is a connection on a certain vector bundle over a base space S of a family of algebraic varieties V_s. The fibers of the vector bundle are the de Rham cohomology groups ${\displaystyle H_{DR}^{k}(V_{s})}$ of the fibers V_s of the family. It was introduced by Manin (1958) for curves S and by Grothendieck (1966) in higher dimensions.

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.

A commonly cited example is the Dwork construction of the Picard–Fuchs equation. Let

Here, ${\displaystyle \lambda }$ 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). Thus, the base space of the bundle is taken to be the projective line. For a fixed ${\displaystyle \lambda }$ in the base space, consider an element ${\displaystyle \omega _{\lambda }}$ of the associated de Rham cohomology group

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