# Rank of a jet bundle of a vector bundle. Interpretation of the first jet bundle

I am trying to understand the jet bundles but currently I am stuck on the following questions:

Let $\pi: E\rightarrow X$ be a smooth (holomorphic) vector bundle of rank $k$ over a smooth (complex) manifold $X$. I know that the bundle $J_k(E)$ of k-jets of $E$ has the structure of a vector bundle over $X$.

I would like to know however:

1.) What is the rank of the vector bundle $J_k(E)$ ?

2.) Is $J_k(E)$ holomorphic in the case when $(E, \pi, X)$ is holomorphic?

3.) When $\pi: E\rightarrow X$ is a fiber bundle with structure group $G$, can we view $J_1(E)$ as the associated principal $G$-bundle $P$ associated to $E$ or not? I have seen an interpretation of $J_1(E)$ as some sort of an "extended frame bundle" of E in the sense that its fiber consists of the set of all pairs comprising a basis of $T_pX$ $(T^{1, 0}_pX)$ and a basis of $E_p$, $p\in X$.

P.S.: I am new here and I really hope that I don't annoy the experienced audience in this forum with trivialities. I would appreciate any help or suggestions or simply good references. Thank you in advance for your competent help.

(1) Locally, jets of sections are just collections of $r=\operatorname{rank}E$ jets of functions, hence, the rank of $J_k(E)$ equals $r$ times the number of multiindices $I=(i_1,\ldots,i_n)$ with $|I|\le k$.

(2) It is certainly holomorphic.

(3) It seems to me that $J_1(E)=T^*X\otimes E$.

• $T^*X \otimes E$ is the kernel of the jet projection $J^1 E \to E$.
– ಠ_ಠ
Nov 10, 2016 at 9:37

Let me just complete Alex' answer to 3). First of all, the jet bundles have nothing to do with any structure group; they are associated to vector bundles, period. Then $J_1(E)$ fits into an exact sequence: $$0\rightarrow \Omega ^1_X\otimes E\rightarrow J_1(E)\,\buildrel {e}\over {\longrightarrow} \,E\rightarrow 0\ .$$ At each point $p\in X$, with maximal ideal $\mathfrak{m}_p\subset \mathcal{O}_{X,p}$, a 1-jet is just a function (in a neighborhood of $p$) modulo those vanishing at order $\geq 2$ at $p$, that is, modulo $\mathfrak{m}_p^2$. The homomorphism $e$ associates to such a function its value at $p$; associating to a function vanishing at $p$ its differential gives the isomorphism of $\mathrm{Ker}( e)$ with $\Omega ^1_X\otimes E$.

This exact sequence plays an important role: its extension class $\mathrm{at} \in \mathrm{Ext}^1_{\mathcal{O}_X}(E,\Omega ^1_X\otimes E)\cong H^1(X,\Omega ^1_X\otimes \mathcal{E}nd(E))$ is the famous Atiyah class. The vanishing of that class is a necessary and sufficient condition for the existence of a section, which is equivalent to the existence of a holomorphic connection on $E$. The Chern classes of $E$ in $\ \oplus\, H^p(X,\Omega ^p_X)$ can be constructed from $\ \mathrm{at}\$ by applying invariant polynomials to $\mathcal{E}nd(E)$.

• Very useful response! Your claim however that jet bundles have nothing to do with any structure group provoked another question in me. Assuming the notation from above, let $V$ be a linear representation of say $G=GL_r(\mathbb{C})$, the structure group of the fiber bundle $E$. Then if $J_1(E)$ is not a principal $G$-bundle, the fiber product $J_1(E)\times_{G}V$ is not well-defined, is it? However, I have encountered such objects in the literature. Mar 21, 2014 at 13:49
• What do you mean they have nothing to do with structure groups? Jet bundles are defined for any submersion and are pretty useful in those contexts as well. May 13, 2020 at 16:10

Please let me add something about the question of holomorphicity:

If $$E$$ is a holomorphic vector bundle, then there is a bundle of jets of holomorphic sections, which is holomorphic, as explained by @hm2020. The bundle of 1-jets of smooth sections of $$E$$, on the other side, does not have a holomorphic structure.

Example: let $$E$$ be a trivial bundle over $$X$$, so that the exact sequence described by @abx splits. Then the jet bundle of smooth sections of $$E$$ is $$(T^*_{\mathbb R}X\otimes_{\mathbb R} E) \oplus E = (\Omega^{1,0}(X)\otimes_{\mathbb C} E) \oplus (\Omega^{0,1}(X)\otimes_{\mathbb C} E) \oplus E~.$$ While $$(\Omega^{1,0}(X)\otimes E)\oplus E$$ has a holomorphic structure and identifies with the bundle of jets of holomorphic sections, the other summand $$\Omega^{0,1}(X)$$ does not have a preferred holomorphic structure.

• Thanks Ben ! I edited my answer Feb 23, 2021 at 15:05
• @NicholasTholozan - When you write $\Omega^1(X)$ for a complex manifold $X$ this usually means the cotangent bundle of $X$, which is a holomorphic vector bundle on $X$. Could you give a precise definition of the bundle $\Omega^1(X)$ mentioned in your post?
– user122276
Feb 23, 2021 at 18:10
• I mean the complexification of the real cotangent space, which splits as $\Omega^{1,0}\oplus \Omega^{0,1}$. I think I have often seen these notations in introductions to Hodge theory. Feb 24, 2021 at 8:45
• When you write $\Omega^1(X)\otimes E \oplus E$ this seems to be a holomorphic vector bundle - you tensor $\Omega^1(X)$ with $E$ which is a holomorphic vector bundle. Can you explain?
– user122276
Feb 25, 2021 at 9:56
• I changed my notation to hopefully less confusing ones. Feb 25, 2021 at 10:47

Question: "1) I would like to know however: What is the rank of the vector bundle $$J^k(E)$$?

1. Is $$J^k(E)$$ holomorphic in the case when $$(E,π,X)$$ is holomorphic?"

Question 2:

Here you find an explicit and elementary construction of the holomorphic jetbundle $$J^k(E)$$:

https://math.stackexchange.com/questions/45627/grothendieck-connections-and-jets/3965791#3965791

Citation: 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.

Example. There is for every $$l\geq 1$$ a short exact sequence

$$0 \rightarrow I^{l}/I^{l+1}\otimes E \rightarrow J^l(E) \rightarrow J^{l-1}(E) \rightarrow 0$$

of locally trivial $$\mathcal{O}_M$$-modules. When you take the associated "holomorphic vector bundle" you should get a short exact sequence of holomorphic vector bundles

$$0 \rightarrow (I^l/I^{l+1})_h \rightarrow J^l_h(E) \rightarrow J^{l-1}_h(E) \rightarrow 0.$$

When $$l=1$$ you should get the "Atiyah sequence"

$$A1.\text{ }0 \rightarrow \Omega^1_h \otimes E_h \rightarrow J^1_h(E) \rightarrow E_h \rightarrow 0.$$

The Atiyah A1 sequence is an exact sequence of holomorphic vector bundles.

Question 1.: If $$rk(E)=e, dim(X)=n$$ it follows there is an explicit formula for the rank:

$$rk(J^k(E))=e\binom{n+k}{n}.$$

Note 1. For a complex projective manifold $$X_{an} \subseteq \mathbb{P}^n_{\mathbb{C}}$$ and a locally free finite rank $$\mathcal{O}_X$$-module $$E$$ on $$X$$, where $$X$$ is the algebraic variety corresponding to $$X_{an}$$, the following holds: There is an isomorphism

$$J^l_{X}(E)_{an} \cong J^l_{X_{an}}(E_{an})$$

of locally trivial $$\mathcal{O}_{X_{an}}$$-modules.

Note 2. If $$X\subseteq \mathbb{P}^m_{\mathbb{C}}$$ is a complex projective variety of dimension $$d$$ and $$E$$ a locally trivial $$\mathcal{O}_X$$-module of rank $$e$$, let $$X(\mathbb{R})$$ be the "underlying" real algebraic variety of $$X$$ (the Weil restriction of $$X$$) and let $$E(\mathbb{R})$$ be the "underlying" locally trivial $$\mathcal{O}_{X(\mathbb{R})}$$-module of rank $$2e$$. It follows

$$W1. \text{ }rk(J^l(E)(\mathbb{R}))=2e\binom{d+l}{d}$$

and

$$W2.\text{ }rk(J^l_{X(\mathbb{R})}(E(\mathbb{R}))=2e\binom{2d+l}{2d}$$

hence these two bundles are not isomorphic in general: The Weil restriction functor does not "commute" with the "jet bundle functor". The bundle in W1 has by definition a structure as $$\mathcal{O}_X$$-module, the bundle in W2 does not.