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dg.differential geometry – Rank of a jet bundle of a vector bundle. Interpretation of the primary jet bundle retort

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dg.differential geometry – Rank of a jet bundle of a vector bundle. Interpretation of the primary jet bundle

**Question: **” I would affection to know however: What is the rank of the vector bundle $J^k(E)$? 2.) Is $J^k(E)$ holomorphic in the illustration when $(E,π,X)$ is holomorphic?”

Question 2:

Here you discover an categorical and elementary development of the holomorphic jetbundle $J^ok(E)$:

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

Citation: occasion. Let $M$ breathe an advanced manifold with construction sheaf $mathcal{O}_M$ and let $mathcal{E}$ breathe a regionally trifling $mathcal{O}_M$-module of finite rank. Let $mathcal{O}_{Mtimes M}$ breathe the construction sheaf of the product manifold and let $p,q:Mtimes M rightarrow M$ breathe the 2 projection maps. You could outline $J^l_M:=mathcal{O}_{Mtimes M}/I^{l+1}$ the place $I$ is the “model of the diagonal”, and $J^l_M(mathcal{E}):=p_*(J^l_M otimes q^*mathcal{E})$.
There is a Taylor morphism,

T6. $T^l: mathcal{E}rightarrow J^l_M(mathcal{E})$

just like the map T1 outlined within the algebraic illustration. level to that when $f:X rightarrow Y$ is a map of sophisticated manifolds, the drag advocate $f^*mathcal{E}$ is outlined as follows. The map $f$ induce a map

T7. $f^{#}:mathcal{O}_Y rightarrow f_*mathcal{O}_X$.

Let $Usubseteq Y$ breathe an launch clique and let $f_U:f^{-1}(U)rightarrow U$ breathe the restricted map. Since $f$ is holomorphic it follows $f_U$ is holomorphic.
If $error mathcal{O}_Y(U)$ is a holomorphic obligation it follows $scirc fin mathcal{O}_X(f^{-1}(U))$ is a holomorphic obligation, inducing the map $f^{#}$. We purchase 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 outline

T8. $f^*mathcal{E}:=mathcal{O}_Xotimes_{f^{-1}(mathcal{O}_Y)}f^{-1}(mathcal{E})$.

It follows the left $mathcal{O}_X$-module $J^l(mathcal{E})$ is a regionally trifling sheaf of finite rank on $X$. I endure Hartshornes relate II.5.18 is correct on this illustration, therefore there may be an equivalence of classes between the class of finite rank regionally trifling sheaves on $X$ and the class of finite rank holomorphic vector bundles on $X$. Hence to $J^l(mathcal{E})$ you must purchase a holomorphic vector bundle $J^l_{h}(mathcal{E})$ whose fiber is the fiber described above. This offers a worldwide definition precise for any finite rank regionally free sheaf on any sophisticated manifold.

Question 1.: If $rk(E)=e, dim(X)=n$ it follows there may be an categorical components for the rank:

$$rk(J^ok(E))=ebinom{n+ok}{n}.$$

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