Grothendieck connections and jets 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? 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.
 A: Beware: I'm not an expert on jet bundles, so the following may not answer your question.
The $k$-jet bundle $J^k V$ of $V$ is defined by 
$$ (J^k V)_x = \mathcal O_x(V) / (\mathfrak m_x^{k+1} \cdot \mathcal O_x(V)) $$
where $\mathfrak m_x$ is the unique maximal ideal of the ring $\mathcal O(V)_x$ (this is at a point $x \in X$). This pointwise definition glues to give a holomorphic vector bundle $J^k V$ over $X$.
From the definition, we see there is a natural map
$$ j^k : \mathcal O(V) \to J^k V $$
defined by passage to the quotient. My guess is that this is the map $j^k$ that Deligne is referring to.
This is of course well and good, but exhibits the trees rather than the forest. You are absolutely right in that there is a link between jets and Taylor series. In fact, morally speaking, the $k$-jet bundle of $V$ is just the bundle whose sections are Taylor developments of order $k$ of sections of $V$. The map $j^1$ that Deligne refers to is thus just the map which sends a section $\sigma$ to its Taylor development of order 1.
It is easiest to see what is going on in local coordinates. Let's suppose that $V$ is a line bundle and look at 1-jets for simplicity -- everything works the same for vector bundles of arbitrary rank and $k$-jets.
Fix a point $x \in X$, and take coordinates $(z_1, \ldots, z_n)$ centered at $x$. Let $e$ be a holomorphic section of $V$ which trivializes $V$ on our coordinate neighborhood. A section $\sigma$ of $V$ may be written as $\sigma(z) = f(z) \, e(z)$, where $f$ is a holomorphic function. The function $f$ has a Taylor development $f(z) = a_0 + a_1 \, z + O(|z^2|)$ around $x = 0$, thus
$$ \sigma(z) = a_0 \, e(z) + a_1 z \, e(z) + O(|z^2|) \, e(z) $$
around $0$. The $1$-jet of $\sigma$ around $x$ is then equal to
$$ j^1(\sigma(z)) = a_0 \, e(z) + a_1 z \, e(z).$$
The $1$-jet bundle $J^1 V$ is thus a rank 2 vector bundle over $X$, and the coefficients $a_0$ and $a_1$ define coordinates along the fibers of $J^1 V$ around $x$.
I don't really know of a good reference for jet bundles. The little I know mostly comes from Chapter VII of Demailly's book (http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf). What I wrote here can be found on pages 351--352 of his book. A little later in the Chapter he proves Kodaira's embedding theorem with jet bundles, so that might interest you.
[Edit:] I just noticed that $j^k$ doesn't seem to be a differential operator. However, if we take $j^1$ to be the quotient map, and make it forget the constant term $a_0$, then it is a differential operator. Maybe this is the map Deligne has in mind?
A: Question: "Thanks for the helpful explanation of the connection with Taylor expansions. I'm still unsatisfied with the definition of j1, partially because this "pointwise definition" of the jet bundle can't be the whole story. After all, one can't in general construct a sheaf just by specifying its stalks. There must be a way to say it with the global definition I gave above: more explicitly, J1(V)=p2∗(OX1⊗OXp−12V)."
Answer. There is an elementary discussion of jets and Taylor expansions of rational functions at the following link:
https://mathoverflow.net/questions/55244/why-must-nilpotent-elements-be-allowed-in-modern-algebraic-geometry/378811#378811
In general if $A\rightarrow B$ is a map of commutative unital rings and if $E$ is any $B$-module, you may define the $l$'th jet bundle $J^l(E):=B\otimes_A B/I^{l+1}\otimes_B E$. There is a canonical map
T1. $T^l: E\rightarrow J^l(E)$
defined by $T^l(e):=1\otimes 1\otimes e$. When $l=1$ you get the map Deligne speaks of in his book. There is when $J^1$ is a finite rank projective $B$-module and $E$ is a finite rank projective $B$-module, a short exact sequence of left $B$-modules
T2. $0 \rightarrow \Omega^1_{B/A}\otimes_B E \rightarrow J^1(E) \rightarrow E \rightarrow 0,$
and the sequence T2 is split iff there is a connection $\nabla: E \rightarrow \Omega^1_{B/A}\otimes_B E$. This gives a "global" definition of the Taylor morphism $T^l$. The Taylor morphism is a differential operator of order $\leq l$:
T3. $T^l\in \operatorname{Diff}^l_A(E, J^l(E))$
for any $l \geq 1$.
If $A:=k$ is a field and $\mathfrak{m}\subseteq B$ corresponds to a $k$-rational point, it follows there is an isomorphism
T4.  $J^l(\mathfrak{m}):= J^l\otimes_{B_{\mathfrak{m}}}\kappa(\mathfrak{m}) \cong  B/\mathfrak{m}^{l+1}$.
More generally $J^l(E)(\mathfrak{m})\cong E/\mathfrak{m}^{l+1}E$, and the Taylor morphism $T^l(\mathfrak{m})$ is the canonical map
T5. $T^l(\mathfrak{m}): E \rightarrow E/\mathfrak{m}^{l+1}E$
defined by $T^l(\mathfrak{m})(e):=\overline{e}$.
Example. Let $M$ be a complex manifold with structure sheaf $\mathcal{O}_M$ and let $\mathcal{E}$ be a locally trivial $\mathcal{O}_M$-module of finite rank. Let $\mathcal{O}_{M\times M}$ be the structure sheaf of the product manifold and let $p,q:M\times M \rightarrow M$ be the two projection maps. You may define $J^l_M:=\mathcal{O}_{M\times M}/I^{l+1}$ where $I$ is the "ideal of the diagonal", and $J^l_M(\mathcal{E}):=p_*(J^l_M \otimes q^*\mathcal{E})$.
There is a Taylor morphism,
T6. $T^l: \mathcal{E}\rightarrow J^l_M(\mathcal{E})$
similar to the map T1 defined in the algebraic case. Note that when $f:X \rightarrow Y$ is a map of complex manifolds, the pull back $f^*\mathcal{E}$ is defined as follows. The map $f$ induce a map
T7. $f^{\#}:\mathcal{O}_Y \rightarrow f_*\mathcal{O}_X$.
Let $U\subseteq Y$ be an open set and let $f_U:f^{-1}(U)\rightarrow U$ be the restricted map. Since $f$ is holomorphic it follows $f_U$ is holomorphic.
If $s\in \mathcal{O}_Y(U)$ is a holomorphic function it follows $s\circ f\in \mathcal{O}_X(f^{-1}(U))$ is a holomorphic function, inducing the map $f^{\#}$. We get an induced map $\tilde{f}: f^{-1}(\mathcal{O}_Y)\rightarrow \mathcal{O}_X$. Since $\mathcal{E}$ is an $\mathcal{O}_Y$-module it follows $f^{-1}(\mathcal{E})$ is an $f^{-1}(\mathcal{O}_Y)$-module, and we define
T8. $f^*\mathcal{E}:=\mathcal{O}_X\otimes_{f^{-1}(\mathcal{O}_Y)}f^{-1}(\mathcal{E})$.
It follows the left $\mathcal{O}_X$-module $J^l(\mathcal{E})$ is a locally trivial sheaf of finite rank on $X$. I believe Hartshornes Exercise II.5.18 is true in this case, hence there is an equivalence of categories between the category of finite rank locally trivial sheaves on $X$ and the category of finite rank holomorphic vector bundles on $X$. Hence to $J^l(\mathcal{E})$ you should get a holomorphic vector bundle $J^l_{h}(\mathcal{E})$ whose fiber is the fiber described above. This gives a global definition valid for any finite rank  locally free sheaf on any complex manifold.
