I start studying some complex geometry and I am lost about some facts about vector bundles.
Consider a smooth manifold $M$; and a holomorphic manifold $X$
I know that in the case of real smooth vector bundles over $M$ we have same canonical isomorphisms mirroring canonical constructions of linear algebra. For example,
- $E^{*}\otimes F\ \simeq \operatorname{Hom}_{\mathbb{R}}(E,F)$
- $(E^{*})^{*} \simeq E$
- $\operatorname{Hom}_{\mathbb{R}}(E_1,F_1)\otimes \operatorname{Hom}_{\mathbb{R}}(E_2,F_2) \simeq \operatorname{Hom}_{\mathbb{R}}(E_1\otimes E_2; F_1 \otimes F_2)$
So in the case of smooth complex vector bundles over $M$; or in the case of holomorphic vector bundles over $X$ could we extend the same isomorphisms from linear algebra to vector bundles?
If this is the case how it is justified?
Is the lack of partitions of unity a problem for this to extend on smooth complex or holomorphic bundles? Because I no that for real smooth bundles one has the isomorphism $E \simeq E^{*}$ but this not true for the smooth complex vector bundles.
Question: "In fact $\Gamma(∙,E)$ is a sheaf on the base space, c1.1 so are the isomorphisms extend to sheaf isomorphisms of $C^{\infty}(M)$-modules in case of real vector bundles? c1.2 and the same for complex vector bundles? c2 Are the isomorphisms extend to sheaf isomorphisms of $\mathcal{O}_X$ -modules in case of holomorphic vector bundles?"
Answer: If $(X, \mathcal{O})$ is a complex manifold there is an "equivalence of categories" between the category of finite rank holomorphic vector bundles on $X$ and the category of locally trivial $\mathcal{O}$-modules hence we may argue using locally free sheaves.
If $(X,\mathcal{O})$ is a locally ringed space and $E$ a finite rank locally trivial $\mathcal{O}$-modules on $X$ and $F$ any $\mathcal{O}$-module, there is a "canonical map"
$$\phi: E^*\otimes_{\mathcal{O}} F \rightarrow Hom_{\mathcal{O}}(E,F)$$
defined as follows: Let $U \subseteq X$ be an open set and let $F(U):=E^*(U) \otimes_{\mathcal{O}(U)}F(U)$ be the presheaf with associated sheaf $E^*\otimes F$.
There is a map
$$\phi(U): F(U) \rightarrow Hom_{\mathcal{O}(U)}(E(U), F(U))$$
defined by
$$\phi(U)(f \otimes w)(v):=f(v)w \in F(U).$$
The map $\phi$ is a map of presheaves and gives rise to a map
$$\phi^+: E^*\otimes F \rightarrow Hom_{\mathcal{O}}(E,F).$$
Since $E^*$ is finite rank locally trivial it follows $\phi^+$ is an isomorphism. You prove this by choosing a local trivialization of $E$.
Example: If $E:=A\{e_1,..,e_n\}$ is a free $A$-module with dual $E^*:=A\{x_1,..,x_n\}$ and $x_i:=e_i^*$ it follows
$$Hom_A(E,F) \cong F^n, E^*\otimes_A F \cong F^n$$
and the canonical map $E^*\otimes_A F \rightarrow Hom_A(E,F)$ induce an isomorphism. Let $\phi: E \rightarrow F$ with $\phi(\sum_j a_je_j):=\sum_ja_j\phi(e_j)$ and define
$$u:= \sum_i x_i \otimes \phi(e_i) \in E^* \otimes F.$$
It follows $u$ maps to $\phi$.
You may do similar constructions for all bundles mentioned above.
Example:
$$Hom(E_1,F_1)\otimes Hom(E_2,F_2)\cong (E_1)^*\otimes F_1 \otimes (E_2)^*\otimes F_2 \cong $$
$$E_1^*\otimes E_2^* \otimes F_1 \otimes F_2 \cong (E_1\otimes E_2)^*\otimes F_1\otimes F_2 \cong Hom(E_1\otimes E_2,F_1\otimes F_2).$$
As mentioned in the comments: Whenever you have a "canonical isomorphism" between vector spaces, such constructions carry over to locally free sheaves.
$$Hom_{\mathcal{O}}(E\otimes F,G) \cong Hom_{\mathcal{O}}(E, F^*\otimes G)$$
etc..
Question: "Is the lack of partitions of unity a problem for this to extend on smooth complex or holomorphic bundles?"
Answer: In the case of holomorphic bundles: By the above argument, no. There is for a real smooth manifold $M$ the "Serre-Swan theorem" saying there is an equivalence of categories between the category of finite rank real smooth vector bundles on $M$ and the category or finite rank projective $C^{\infty}(M)$-modules. Hence you may prove similar results using homological algebra. The formula
$$Hom_A(E\otimes F,G) \cong Hom_A(E,Hom_A(F,G)$$
holds for any $A$-modules $E,F,G$ with $E$ a (finite rank) projective $A$-modules. This proves the same result for vector bundles on $M$.
There is the "list of canonical isomorphisms in homological algebra and sheaf theory":
Stalks of exterior power
The "magic diagram" is cartesian
Prove $\phi$ to be isomorphism (an exercise in commutative algebra)
Form of basic open set of affine scheme: The intersection of two basic open sets.