Grothendieck connection

In algebraic geometry and synthetic differential geometry, a Grothendieck connection is a way of viewing connections in terms of descent data from infinitesimal neighbourhoods of the diagonal.

Introduction and motivation

The Grothendieck connection is a generalization of the Gauss-Manin connection constructed in a manner analogous to that in which the Ehresmann connection generalizes the Koszul connection. The construction itself must satisfy a requirement of "geometric invariance", which may be regarded as the analog of covariance for a wider class of structures including the schemes of algebraic geometry. Thus the connection in a certain sense must live in a natural sheaf on a Grothendieck topology. In this section, we discuss how to describe an Ehresmann connection in sheaf-theoretic terms as a Grothendieck connection.

Let "M" be a manifold and π : "E" → "M" a surjective submersion, so that "E" is a manifold fibred over "M". Let J¹("M","E") be the first-order jet bundle of sections of "E". This may be regarded as a bundle over "M" or a bundle over the total space of "E". With the latter interpretation, an Ehresmann connection is a section of the bundle (over "E") J¹("M","E") → "E". The problem is thus to obtain an intrinsic description of the sheaf of sections of this vector bundle.

Grothendieck's solution is to consider the diagonal embedding Δ : "M" → "M" × "M". The sheaf "I" of ideals of Δ in "M" × "M" consists of functions on "M" × "M" which vanish along the diagonal. Much of the infinitesimal geometry of "M" can be realized in terms of "I". For instance, Δ^* ("I"/"I"²) is the sheaf of sections of the cotangent bundle. One may define a "first-order infinitesimal neighborhood" "M"⁽²⁾ of Δ in "M" × "M" to be the subscheme corresponding to the sheaf of ideals "I"². (See below for a coordinate description.)

There are a pair of projections "p"₁, "p"₂ : "M" × "M" → "M" given by projection the respective factors of the Cartesian product, which restrict to give projections "p"₁, "p"₂ : "M"⁽²⁾ → "M". One may now form the pullback of the fibre space "E" along one or the other of "p"₁ or "p"₂. In general, there is no canonical way to identify "p"₁^*"E" and "p"₂^*"E" with each other. A Grothendieck connection is a specified isomorphism between these two spaces.

References

# Osserman, B., "Connections, curvature, and p-curvature", "preprint".
# Katz, N., "Nilpotent connections and the monodromy theorem", "IHES Publ. Math." 39 (1970) 175-232.