Grothendieck connections and jets

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?