Gauss-Manin connection

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In mathematics, the Gauss-Manin connection is the connection on a vector bundle over a family of algebraic varieties. The base space is taken to be the set of parameters defining the family, and the fibres are taken to be the de Rham cohomology group H^k_{DR}(V) of the variety 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 fibre in the family to nearby fibres, 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

V_\lambda(x,y,z) = x^3+y^3+z^3 - \lambda xyz=0 \;

be the projective variety describing the elliptic curve. Here, λ is a free parameter describing the curve; it is an element of the complex projective line. Thus, the base space of the bundle is taken to be the projective line. For a fixed λ in the base space, consider an element ωλ of the associated de Rham cohomology group

\omega_\lambda \in H^1_{Dr}(V_\lambda)

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

(\lambda^3-27) \frac{\partial^2 \omega_\lambda}{\partial \lambda^2} +3\lambda^2 \frac{\partial \omega_\lambda}{\partial \lambda} + \lambda \omega_\lambda =0

In the more abstract setting of D-module theory, the existence of such equations is subsumed in a general discussion of the direct image.

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