# 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 '16 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. – Flavius Aetius Mar 21 '14 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. – Bananeen May 13 '20 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 – Nicolas Tholozan Feb 23 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? – hm2020 Feb 23 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. – Nicolas Tholozan 2 days ago
• 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? – hm2020 yesterday
• I changed my notation to hopefully less confusing ones. – Nicolas Tholozan yesterday

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.