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(page 17)

...It follows that the projection (of tangent spaces, respectively real, complexified and holomorphic, at $p$ to a complex manifold $M$)

$$T_{\mathbb{R},p}(M)\longrightarrow T_{\mathbb{C},p}(M)\longrightarrow T_{p}'(M) $$

is an $\mathbb{R}$-linear isomorphism. This last feature allows us to "do geometry" purely in the holomorphic tangent space. For example, let $z(t) = x(t) + iy(t)$, and the tangent to the arc may be taken either as $$x'(t)\frac{\partial}{\partial x}+ y'(t)\frac{\partial}{\partial y}$$ in $T_{\mathbb{R},p}(\mathbb{C})$ or

$$z'(t)\frac{\partial}{\partial z}$$ in $T'(\mathbb{C})$ and these two correspond under the projection.

What does the book mean by "doing geometry"?

I guess more basically, what properties precisely do $\mathbb{R}$-linear isomorphisms preserve that are fundamental to 'doing geometry'? I assume only angles?

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up vote 7 down vote accepted

($M$ is a complex manifold of complex dimension $n$.) The space $T_{\mathbb{R},p}(M)$ is the real tangent space of the real ($2n$ dimensional) manifold $M$. But what you really want is a complex tangent space of dimension $n$. First, they consider $T_{\mathbb{C},p}(M) = T_{\mathbb{R},p}(M) \otimes_{\mathbb{R}} \mathbb{C}$, which is a complex vector space, but of dimension $2n$. This vector space splits nicely into $T_{\mathbb{C},p}(M) = T'_p(M) \oplus T''_p(M)$. Here $T'_p(M)$ is the subspace of derivations that map all antiholomorphic functions to $0$. On the other hand, $T''_p(M)$ kills all holomorphic functions, so those derivations aren't very interesting (since you'll be studying holomorphic functions on $M$). That's why $T'_p(M)$ is the natural choice of $n$-dimensional complex vector space to call the (complex) tangent space to $M$ at $p$.

You're putting the emphasis on the wrong part of the bolded sentence. "Do geometry" just means using the usual techniques to analyze a space, including considering tangent spaces, the tangent bundle, sections of the tangent bundle, exterior powers of the tangent bundle, etc. The key part of the sentence is that you will be working "purely in the holomorphic tangent space." The $\mathbb{C}$-vector space $T'_p(M)$ carries more information (the complex vector space structure) than the $\mathbb{R}$-vector space $T_{\mathbb{R},p}(M)$. In fact, the first application of this fact is found in the end of this subsection, where they prove that every complex manifold has a canonical orientation (which is absolutely not true for real manifolds).

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