# 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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