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In his book, Daniel Huybrechts define the complexified tangent bundle as: $T_{\mathbb C}U:= TU \otimes \mathbb C$.

But I don't understand this tensor product, I understand what $T_xU \otimes \mathbb C$ is, but here $TU$ is a vector bundle on $U$, while $\mathbb C$ is not, but the definition of tensor product between vector bundles I found on Wikipedia require that the both vector bundles are over the same space.

Can anyone help me please ?

Johny06
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    This is the vector bundle whose fibers are $T_x(U) \otimes \mathbb{C}$. Alternatively, it's the tensor product with the trivial vector bundle with fibers $\mathbb{C}$. – Qiaochu Yuan Sep 20 '22 at 17:55
  • Okay thanks, that is what I guessed, but because $\mathbb C$ is also a vector space I was afraid to miss something. – Johny06 Sep 20 '22 at 18:06
  • $\mathbb{C}$ is a real vector space of dimension $2$. Do you know why? – Armando j18eos Sep 23 '22 at 20:44
  • What is your point Armando ? I was confused because my definitions of tensor product was between vector spaces or between vector bundle, while here $TU$ is a vector bundle on $U$ but $\mathbb C$ is not – Johny06 Sep 25 '22 at 12:09

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