Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

1 Answer 1

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?

share|improve this answer
    
Thanks for the helpful explanation of the connection with Taylor expansions. I'm still unsatisfied with the definition of $j^1$, 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, $J^1(V) = p_{2*}(\mathcal{O}_{X_1} \otimes_{\mathcal{O}_X} p_2^{-1}V)$. –  Justin Campbell Jun 16 '11 at 22:02
1  
Yes, we still have to say how glue the local trivializations of $J^k V$, but its transition morphisms of are induced by those of $V$. I'm afraid I can't help much with the algebraic definition as I'm just a poor differential geometer, not used to the more sophisticated ways of the world. We're more the "Hulk! Smash!" type of people. –  Gunnar Magnusson Jun 17 '11 at 7:34

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.