Let $(E,\pi,M)$ be a holomorphic bundle, i.e. $(M,J)$ is a complex manifold and $\pi \colon E \to M$ is a complex bundle such that there exists a trivialization with holomorphic transition functions.
I am asked to prove that the dual vector bundle $E^* \to M$ is also holomorphic.
I am pretty sure that this is the kind of problem that should be solved looking at how transition maps change, i.e. I should probably define the dual bundle as the one having $g_{\alpha,\beta}^*$ as its transition functions (whatever this means) and then the result follows from the fact that a bundle can be reconstructed from its transitions functions… What I need is to express the $g_{\alpha,\beta}^*$ with the $g_{\alpha,\beta}$ to obtain that this new bundle is holomorphic.
Thank you for your help!!!
EDIT: Here's my attempt: suggestions, corrections and improvements are encouraged!!!
1) I define a fiber bundle $F$ with the fiber bundle construction theorem (http://en.wikipedia.org/wiki/Associated_bundle) as the one having $(g_{\alpha,\beta}^t)^{-1}$. This is clearly a holomorphic vector bundle, being $g_{\alpha,\beta}$ holomorphic.
2) I want to prove that this corresponds to the idea of fiber bundle given in class: I want to prove that $F_p \cong (E_p)^*$. For this purpose I would like to define a duality between $F_p$ and $E_p$, i.e. $E_p \times F_p \to \mathbb{C}$.
Let $\{(U_{\alpha},\psi_{\alpha})\}$ be a trivializzation of $E$, and let $\{(U_{\alpha},\psi^*_{\alpha})\}$ be a trivializzation of $F$.
If $p \in U_{\alpha}$, then
$$E_p \times F_p \to \mathbb{C}$$
$$((p,v), (p,w)) \mapsto \langle \pi_2\psi_{\alpha}((p,v)),\pi_2\psi^*_{\alpha}((p,w)) \rangle$$
Here $\pi_2$ is the projection $\pi_2 \colon U_{\alpha} \times \mathbb{C}^k \to C^{k}$, and $\langle \cdot,\cdot \rangle$ is the standard Hermitian product on $\mathbb{C}^k$.
I would like also to prove this application is well defined, i.e., if $p \in U_{\alpha} \cap U_{\beta}$ I would like this to be independent of $\alpha$. Here is where I am supposed to show that the choise of the transition functions for $F$ is what makes things work. (At the moment, I am stuck with this, help please!!!)
The application $v \to \langle \pi_2\psi_{\alpha}((p,v)),\pi_2\psi^*_{\alpha}((p,w)) \rangle$ seems to be a linear application from $E_p \to \mathbb{C}$ and this should give the result I was looking for. Am I correct?
Best Answer
I am contemplating about the same thing right now. The cocycle construction of dual bundle is easy for definition but it does not lie in my head solidly. The following link seems to have some clues about how to construct it in terms taking duals of fibers of a given vector bundle. I will update my answer if I figure out something more.
http://www.math.sunysb.edu/~azinger/mat531-spr10/vectorbundles.pdf
Update: I have edited my answer in great detail, but it somehow was rejected by the website. I believe the answer to your question is just a matter of which definition you take, and whichever definition you take, the holomorphy of $E^{*}$ is given by that of $E$ because the charts of $E^{*}$ is induced by local biholomorphic maps $E(U) \simeq E^{*}(U)$ given in the data of local trivialiaztion. $E^{*}$ is defined to be holomorphic, which is funny that one should prove it, if I understood correctly.