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

Question

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.

Answer 1

(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 $J_1(E)=T^*X\otimes E$.

Answer 2

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 $\mathcal{E}nd(E)$.

Answer 3

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.

Answer 4

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

2. 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.