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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?

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2 Answers 2

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comes a time in every man's life when the only thing left is its partial solution to the problem. I am sorry but I am not going to invest more reputation for this question. I am of course interested in the answer, so if you want to post something, this is always wellcome.

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.

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if you go to local coordinates everything becomes as in the world of vector spaces. There, the dual map of a matrix is the transpose. So it is for vector bundles. In The book of Huybrechts, "complex geometry" sec. 2.2. p.66 there is something written more about it.

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