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The following question is based on some remarks in section I.2 of Deligne's book Equations Différentielles à Points Singuliers Réguliers.

Let $X$ be a smooth complex variety and $X_1$ the first infinitesimal neighborhood of the diagonal in $X \times X$, so there is a natural morphism $X \to X_1$. If we write $p_1,p_2 : X_1 \to X$ for the two projections, then the first-order jet bundle of a vector bundle $V$ on $X$ is defined to be $J^1(V) = p_{1*} p_2^*V$. Here "upper star" is used in the sense of $\mathcal{O}$-modules. This allows for a convenient way of expressing the notion of a connection: this is just an isomorphism $p_1^*V \to p_2^*V$ which restricts to the identity over $X$, which is the same as an $\mathcal{O}$-linear map $V \to J^1(V)$ such that the composition $V \to J^1(V) \to V$ is the identity.

Here Deligne says something I don't understand: he refers to a first-order differential operator $j^1 : V \to J^1(V)$ "which associates with any section a first-order jet" (I'm translating from the French). What is he talking about, and what makes the map he's talking about a differential operator? I don't have much intuition for jets, although I know they have something to do with taking Taylor expansions, so any general explanation of that would be greatly appreciated as well.

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According to EGA IV, section 16.7 we may denote 1-jets by $\mathcal{P}^1_X(V)$ (In fact you may even take for $V$ a coherent sheaf). There is a canonical morphism $V \to\mathcal{P}^1_X(V)$. Notice that it is not $\mathcal{O}$-linear but just $\mathbb{C}$-linear. Now Proposition 16.8.4 tells us $$ \mathcal{H}om_X(\mathcal{P}^1_X(V), W) = \mathcal{D}iff_X(V,W) $$ where $\mathcal{D}iff_X(V,W)$ denotes the module of (first order) differential operators from $V$ to $W$. And the map is induced by precomposing with the canonical map $V \to\mathcal{P}^1_X(V)$ which plays the role of "universal first order differential operator" taking values in $V$. In EGA everything is slightly more general but you may stick to order 1 and all the ideas are already there.

As for the meaning of jets, I suggest to look at the mentioned chapter in EGA, together with the corresponding part in chap. 0 of EGA IV. If differentials are first order approximation, then first order differential operators are approximations up to first order, i.e. preserving the value of the function, not just the slope of the tangent.

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Here's one way to say what's going on.

For a vector bundle $V$ on $X$ and a point $x\in X$, two local sections of $V$ are said to have the same $1$-jet if their Taylor series have the same constant and linear terms. To make sense of "Taylor series" you might need a coordinate chart in $X$ and a local basis for $V$, but this equivalence relation is independent of those choices. The equivalence classes form a vector space. Call it $J^1_x(V)$ and write $j^1_x(v)$ for the class of the section $v$. Varying $x$ now, this yields a vector bundle $J^1(V)$ on $X$. By remembering only the constant term of the Taylor series we get a surjective map of vector bundles $J^1(V)\to V$. The kernel is canonically isomorphic to $T^\star X\otimes V$.

A connection may be defined as a splitting of this, in other words a vector bundle map $V\to J^1(V)$ such that the composed map $V\to V$ is the identity. In the case of the trivial line bundle, where section means function, there is such a splitting, defined using $1$-jets of constant functions. In the general case there is no canonical splitting. A connection may be informally thought of as a choice of which ($1$-jets of) sections of $V$ are being considered locally constant. (Locally connections exist but are not canonical. Globally they need not exist at all in algebraic geometry, although they do in smooth topology.)

There is in any case a canonical map $j^1:\Gamma(V)\to \Gamma(J^1(V))$ of section spaces (or sheaves) that splits that surjection. It takes $v$ to $x\mapsto j^1_x(v)$. But it is not $\mathcal O$-linear, i.e. it is not an order zero operator. The quantity $j^1(fv)-fj^1(v)$, which is a section of that kernel, may be written as $df\otimes v$.

A connection yields an $\mathcal O$-linear map $\Gamma(V)\to \Gamma(J^1(V))$ that splits the surjection. Subtracting it from $j^1$ we get a map $\nabla:\Gamma(V)\to \Gamma(T^\star X\otimes V)$, a first order operator satisfying $\nabla (fv)=f\nabla (v)+df\otimes v$, namely the covariant derivative associated with the connection. Of course the connection can be recovered from $\nabla$.

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Great answer! Do you have some reference for this? Many thanks. – Filippo Alberto Edoardo Jun 14 '12 at 4:17
No, sorry, I'm afraid this is a combination of something I picked up years ago (from lectures, not from a book) and my own thoughts. – Tom Goodwillie Jun 14 '12 at 19:12

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