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Suppose that $E$ is a complex bundle already. This is to say that is can be viewed as a real bundle with an almost complex structure $j$. Then Taubes assert $E$ sits inside its complexification $E_{\mathbb{C}}$ as a complex rank $n$ subbundle.

His construct is as follows: We observe the endomorphism $j$ of $E$ acts as an endomorphism of $E_{\mathbb{C}}$ sending any given $\sigma=\sum_{k}z_{k}\otimes e_{k}$ to $j\sigma=\sum_{k}z_{k}\otimes je_{k}$. As an endomorphism now of $E_{\mathbb{C}}$, this $j$ obeys $j^{2}\sigma=-\sigma$, and it also commutes with multiplication by elements in $\mathbb{C}$. This understood, the bundle $E$ sits inside the bundle $E_{\mathbb{C}}$ as the set of vectors $v$ such that $jv=iv$. This inclusion of $E$ into $E_{\mathbb{C}}$ as a direct summand sends any given vector $e\in E$ to the vector $\frac{1}{2}(1\otimes e-i\otimes e)$.

I am confused because the two operations he introduced - $j$ inherited from $E$ and $i$ inherited from $E_{\mathbb{C}}$ by identifying $i\sum z_{k}\otimes e_{k}=j_{0}z_{k}\otimes e_{k}$. Here $j_{0}$ standards for almost complex structure matrix on $\mathbb{R}^{2}$. Then if $z_{k}=(x_{k},y_{k})$ we have $j_{0}(x_{k},y_{k})=(-y_{k},x_{k})$. But why identify $E$ as the set $jv=iv$? This feels odd to me for the action of $j_{0}$ on $z_{k}$ does not extend to $e_{k}$, while $j$'s action is on $e_{k}$ such that $j^{2}=-1$. If we equate the two sides then we would have $$iz_{k}\otimes e_{k}=z_{k}\otimes je_{k}\Leftrightarrow z_{k}\otimes e_{k}=-iz_{k}\otimes je_{k}=-z_{k}\otimes jje_{k}=z_{k}\otimes e_{k}$$ because $i: e_{k}\rightarrow je_{k}$ is part of the almost complex structure on $E$. However I still do not know how to deduct that this maps $$e\rightarrow \frac{1}{2}(1\otimes e-i\otimes e)$$

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I think that must be a typo: $e$ should be sent to $\frac{1}{2}(1 \otimes e - i \otimes Je)$. – Eric O. Korman Sep 14 '12 at 12:55
you mean $je$? ($j$ from the complex structure on $E$?) – Bombyx mori Sep 14 '12 at 20:21
I fixed a typo, just in case you are confused. – Bombyx mori Sep 14 '12 at 20:24

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