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 . The fibers of the vector bundle are the de Rham cohomology groups of the fibers of the family. It was introduced by Yuri Manin (1958) for curves S and by Alexander 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.
Intuition
Consider a smooth morphism of schemes over characteristic 0. If we consider these spaces as complex analytic spaces, then the
Consider a cohomology class such that where is the inclusion map. Then, if we consider the classes
eventually there will be a relation between them, called the Picard–Fuchs equation. The Gauss–Manin connection is a tool which encodes this information into a connection on the flat vector bundle on constructed from the .[1]
Example
A commonly cited example is the Dwork construction of the Picard–Fuchs equation. Let
- be the elliptic curve .
Here, is a free parameter describing the curve; it is an element of the
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
D-module explanation
In the more abstract setting of
Equations "arising from geometry"
The whole class of Gauss–Manin connections has been used to try to formulate the concept of differential equations that "arise from geometry". In connection with the
See also
References
- ^ "Reference for Gauss–Manin Connection". math.stackexchange.com.
- ^
MR 2641188.
- ^ Reiter, Stefan (2002). "On applications of Katz' middle convolution functor (Deformation of differential equations and asymptotic analysis)" (PDF). Kyoto University Research Information Repository.
- MR 2338364.
- Kulikov, Valentine (1998), Mixed Hodge Structures and Singularities, Cambridge Tracts in Mathematics, pp. 1–59 (Gives and excellent introduction to Gauss–Manin connections)
- Dimca, Alexandru, Sheaves in Topology, pp. 55–57, 206–207 (Gives example of Gauss–Manin connections and their relation to D-module theory and the Riemmann-Hilbert correspondence)
- Griffiths, Phillip, Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems (Gives a quick sketch of main structure theorem of Gauss–Manin connections)
- Barrientos, Ivan, The Gauss-Manin connection and regular singular points. (PDF)
- S2CID 123434721
- "Gauss-Manin connection", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- MR 0103889