TITLE: Deformations of sheaves via automorphisms. How to express $Ext^1$?
QUESTION [5 upvotes]: Let $X$ be a complex manifold (for example $\mathbb CP^n$), let $v$ be a holomorphic vector field on $X$, and let $F$ be a coherent sheaf (for example a vector bundle or a structure sheaf of a point). Then $v$ defines an element in $Ext^1(F,F)$. Indeed $v$ generates an action of $\mathbb C$ on $X$ and taking pull-backs of $F$ under this action we get a deformation of $F$, hence an element of $Ext^1(F,F)$.
Question. Is there any fancy (or not fancy) way to express the corresponding element of $Ext^1(F,F)$ in terms of $v$ and $F$? Maybe there is some construction with jets? (I understand, this is a bit vague)
Added. Two equally nice and far leading answers were given to this question. I would like summarise here what I understood from David's answer in down to earth terms. So, we want to associate an extension  $F\to E\to F$ to a vector field $v$. Suppose (just for the sake of been very much down to earth), that $F$ is a vector bundle, and $v$ has no zeros. Then we can consider $1$-jets of sections of $F$ in the direction of $v$ (or if you want, along trajectories of $v$). It is not hard to see that this is a bundle of rank $2rank(F)$, and this is exactly $E$, that we are looking for.

REPLY [7 votes]: Here is a not so fancy description.
There is a general principle (in algebraic geometry but applicable to some
neighbouring disciplines) that says that anything that is functorial and
commutes with base change admits an action by vector fields. The reason is that
a vector field on $X$ gives an automorphism on $X[\delta]$, the base extension
of $X$ to the ring of dual numbers, that is the identity modulo $\delta$. 
As an example that is unrelated to the question but illustrative take the
example of a vector bundle $E$ and its symmetric algebra $S^*E$. The fact that
the symmetric algebra is defined by a universal property and commutes with base
extension immediately gives an action of derivations of $\mathcal O_X$ on $S^*E$
(the usual proof of this uses these properties implicitly but is less
transparent as it doesn't use them explicitly).
In any case turning to the problem at hand, given a vector field $v$ we get an
automorphism $\varphi$ of $X[\delta]$ and hence an $\mathcal
O_{X[\delta]}$-module which is the pullback by $\varphi$ of the constant
extension $F[\delta]$ of $F$ to $X[\delta]$. Concretely it means that we
consider $F[\delta]=F\bigoplus F\delta$ with action of $f\in\mathcal O_X$ given
by $f\cdot e=fe+v(f)e\delta$ (and $\delta$ acting in the obvious way). The
identification of first order deformations of $F$ with $\mathrm{Ext}^1(F,F)$
is the done by considering this as an extension of $F=F\delta$ by
$F[\delta]/F\delta=F$.
This can be elaborated as follows (which should give a closer relation with the
answer provided by David): Instead of taking the pullback of $F[\delta]$ by
$\varphi$ we can instead consider the vector field as a morphism $\varphi\colon
X[\delta]\rightarrow X$ and pullback $F$ along it. We of course also have the
constant morphism $X[\delta]\rightarrow X$ and the coincide on the closed
immersion $X\hookrightarrow X[\delta]$. Hence the induced map
$X[\delta]\rightarrow X\times X$ factors through the first order neighbourhood
of the diagonal $X^{(2)}:=\mathrm{Spec}\mathcal O_{X\times X}/\mathcal I^2_\Delta
\hookrightarrow X\times X$, i.e., a map $X[\delta]X^{(2)}$ such that the
composite of the first projection $X^{(2)}\rightarrow X$ is $\varphi$ and with
the second is the constant map. (Concretely, we have $\mathcal I_\Delta/\mathcal
I^2_\Delta=\Omega^1_X$ and $v$ is a map $\Omega^1_X\rightarrow\mathcal O_X$ and
the map is just the pushout along this map.) Hence the pullback of $F$ along $\varphi\colon
X[\delta]\rightarrow X$ is obtained by first taking the pullback $F'$ of $F$ along
the first projection $X^{(2)}\rightarrow X$ and the pullback further on to
$X[\delta]$. However, $F'$ as an extension $F\bigotimes\Omega^1_X$ by $F$ is the
Atiyah extension and we the extension of $F$ by $F$ associated to $v$ is just
the pushout of it along the map $F\bigotimes\Omega^1_X \rightarrow F$ induced by
$v$. Hence, the extension is the pushout of the universal Atiyah extension.