What does $T_z\mathbb{R}^2\otimes\mathbb{C}$ in p. 2 of Huybrechts' book mean?

Question

I apologize for my lack of imagination and the likely silliness of this question, but what does $T_z\mathbb{R}^2\otimes\mathbb{C}$ mean here (last paragraph)?

And how does that extension work?

Thank you.

Answer 1

$T_z \Bbb R^2 \otimes \Bbb C$ is the complexification of the real vector space $T_z \Bbb R^2$. Essentially, this operation just changes the field of scalars from $\Bbb R$ to $\Bbb C$, where the complex multiplication is defined on simple tensors by $$z(v \otimes w) = v \otimes (zw)$$ for $v \in T_z \Bbb R^2$ and $z, w \in \Bbb C$.

A linear map $$T: U \longrightarrow V$$ between real vector spaces extends to a linear operator $$T_{\Bbb C}: U \otimes \Bbb C \longrightarrow V \otimes \Bbb C$$ defined on simple tensors by $$T_{\Bbb C}(u \otimes z) = T(u) \otimes z$$ for $u \in U$ and $z \in \Bbb C$.