algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

and

synthetic differential geometry In mathematics, synthetic differential geometry is a formalization of the theory of differential geometry in the language of topos theory. There are several insights that allow for such a reformulation. The first is that most of the analytic dat ...

, a Grothendieck connection is a way of viewing

connections Connections may refer to: * Connection (disambiguation), plural form Television * '' Connections: An Investigation into Organized Crime in Canada'', a documentary television series * ''Connections'' (British TV series), a 1978 documentary tele ...

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 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 H^k_(V_s) of the fibers V_s ...

constructed in a manner analogous to that in which the

Ehresmann connection In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of the notion of a connection, which makes sense on any smooth fiber bundle. In particular, it does ...

generalizes the Koszul connection. The construction itself must satisfy a requirement of ''geometric invariance'', which may be regarded as the analog of

covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables. If greater values of one ...

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 Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics) In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open s ...

on a

Grothendieck topology In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is ca ...

. In this section, we discuss how to describe an Ehresmann connection in sheaf-theoretic terms as a Grothendieck connection. Let

M

be a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

and

\pi : E \to M

surjective In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...

submersion Submersion may refer to: *Being or going underwater, as via submarine, underwater diving, or scuba diving *Submersion (coastal management), the sustainable cyclic portion of foreshore erosion *Submersion (mathematics) * Submersion (Stargate Atlanti ...

, so that

E

is a manifold fibred over

M.

Let

J^1(M, E)

be the first-order

jet bundle In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. ...

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

)

J^1(M, E) \to 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

\Delta : M \to M \times M.

The sheaf

I

of ideals of

\Delta

M \times M

consists of functions on

M \times M

which vanish along the diagonal. Much of the infinitesimal geometry of

M

can be realized in terms of

I.

For instance,

\Delta^*\left(I, I^2\right)

is the sheaf of sections of the

cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This m ...

. One may define a ''first-order infinitesimal neighborhood''

M^

\Delta

M \times M

to be the

subscheme This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. ...

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 \times M \to M

given by projection the respective factors of the Cartesian product, which restrict to give projections

p_1, p_2 : M^ \to M.

One may now form the

pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: ...

of the fibre space

E

along one or the other of

p_1

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. One may proceed to define

curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...

and p-curvature of a connection in the same language.

References

# Osserman, B., "Connections, curvature, and p-curvature", ''preprint''. # Katz, N., "Nilpotent connections and the monodromy theorem", ''IHES Publ. Math.'' 39 (1970) 175–232. {{Manifolds Connection (mathematics) Algebraic geometry