What exactly is the relationship between an Ehresmann connection and splitting of the jet sequence?

Question

An Ehresmann connection on a vector bundle $\pi : E \to X$ is a splitting of the sequence,

$$ 0 \to V \to TE \to \pi^* TX \to 0 $$ which respects the linear structure on $E$ (meaning the section is invariant under the induced automorphism of $T E$ induced by scaling).

In algebraic geometry, I am familiar with the equivalence between connections on a locally free sheaf $\mathcal{E}$ and splitting of the sequence

$$ 0 \to \Omega_X^1 \otimes \mathcal{E} \to J^1(\mathcal{E}) \to \mathcal{E} \to 0. $$ I have heard it said that this is a version of the Ehresmann connection formalism but I am not able to make this precise. If I dualize the top sequence and use the fact that $V \cong \pi^* E$ then I recover,

$$ 0 \to \pi^* T^* X \to T^* E \to \pi^* E^* \to 0 $$ which looks similar to the jet bundle sequence. However, I am not sure how to directly compare these two sequences.

Furthermore, the notion of an Ehresmann connection makes perfect sense in the algebraic category. However, (at least without directly relating it to the jet bundle sequence) I do not see how to show that the datum of a splitting recovers an (algebraic) connection.

The usual construction of a connection from an Ehresmann connection goes through algebraically. Call the splitting $v : T E \to \pi^* E$. Then given a section $s : X \to E$ we get $\mathrm{d}{s} : T X \to s^* T E$ and then $s^* v \circ \mathrm{d}{s}$ is a linear map $T X \to E$ defining $X \mapsto \nabla_X s$ thus defining the connection.

However, to reverse this process, it seems that I need to be able to choose, locally, flat sections for a connection $\nabla$ in order to define the kernel of $v$ which is the horizontal subspace. Does this mean the Ehresmann connection is really a transcendental object?

Answer 1

$\require{AMScd}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\id{id}$No, Ehresmann connection is also algebraic. I think that this should have been discussed in the literature. We consider the algebraic case, which also works for differential geometry but some terminologies of synthetic differential geometry might be necessary (cf. Kock's work).

We fix the following convention: locally free sheaves $\mathcal E$ on $X$ corresponds to vector bundles $\Spec_X(\Sym_{\mathcal O_X}(\mathcal E^\vee))\to X$, thus this correspondence is contravariant.

We first analyze the sections of the canonical map $TE\to\pi^*(TX)$. For every ring $R$, the set of $R$-points of $TX$ is given by maps $\Spec(R[\epsilon])\to X$ where $\epsilon^2=0$. It follows that the set of $R$-points of $\pi^*(TX)$ is given by commutative diagrams

\begin{CD} \Spec(R)@>>>E\\ @VVV@VVV\\ \Spec(R[\epsilon])@>>>X \end{CD}

and lifts of these points to $TE$ correspond to lifts $\Spec(R[\epsilon])\to E$ filling into this diagram, and an Ehresmann connection corresponds to a functorial lift satisfying Zariski descent (with respect to $R$).

As usual, consider the diagonal map $\Delta\colon X\to X\times X$, and let $\mathcal I$ be its ideal sheaf (if $X$ is not separated, we should take an open neighborhood of the diagonal), and $Y:=V(\mathcal I^2)\subseteq X\times X$ the first order thickening of the diagonal. Then we have two projections $p_1,p_2\colon Y\to X$ which share a common section $\Delta\colon X\to Y$ induced by the diagonal map.

Since $p_1\circ\Delta=\id_X$, we get the commutative diagram

\begin{CD} E@>>>p_1^*(E)@>>>E\\ @VVV@VVV@VVV\\ X@>\Delta>>Y@>p_1>>X \end{CD}

where all squares are Cartesian. Composing the middle vertical map with $p_2\colon Y\to X$, we get a map $E\to p_1^*(E)$ over $X$ (Warning: $p_1^*(E)\to X$ is induced by $p_2$, not $p_1$), and connections on $E$ correspond to its right inverses $p_1^*(E)\to E$ over $X$. Equivalently, there is a commutative diagram

\begin{CD} E@=E\\ @VVV@VVV\\ p_1^*(E)@>>>X \end{CD}

and connections on $E$ correspond to lifts $p_1^*(E)\to E$ filling into this diagram. Note that $\mathcal O_{p_1^*(E)}$ is a trivial square-zero extension of $\mathcal O_E$ by a locally free sheaf, thus a section of $TE\to\pi^*(TX)$ gives rise to a connection.

Remark 1. The above argument generalizes to smooth fiber bundles $E\to X$ over a smooth scheme $X$ over a field, or $G$-torsors for smooth $X$-group schemes $G$.

Remark 2. One way to understand the algebraicity is that, for every point $x\in X$, every tangent vector $v\in T_xX$, every connection $\nabla$ on $E$, and every local section $s$ of $E$ around $x$, the value of $\nabla_v(s)$ at $x$ only depends on the "first derivative" of $s$ at $x$ (this is precisely what the Atiyah sequence involving the first jet bundle tells us).