I started studying the book of Daniel Huybrechts, Complex Geometry An Introduction. I tried studying backwards as much as possible, but I have been stuck on the concepts of almost complex structures and complexification. I have studied several books and articles on the matter including ones by Keith Conrad, Jordan Bell, Gregory W. Moore, Steven Roman, Suetin, Kostrikin and Mainin, Gauthier.
I have several questions on the concepts of almost complex structures and complexification. Here is one:
Question: Below, I describe what I understand is happening in Suetin, Kostrikin and Mainin (12.13 of Part I), where the authors prove for any $\mathbb C$-vector space $L$, $L_{\mathbb R}^{\mathbb C} \cong L \bigoplus \overline L$ (I also asked about this isomorphism here.) Also, I base my understanding on Daniel Huybrechts, Complex Geometry An Introduction (Chapter 1.2). Is my understanding, given in the two parts as follows, correct?
Part 0. Assumptions, definitions, notations:
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Complex structure on map definition : See here.
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On $L_{\mathbb R}^2$, we have almost complex structure $J(l,m):=(-m,l)$. $J$ is canonical in the sense that we define complexification $(L_{\mathbb R})^{\mathbb C}$ of $L_{\mathbb R}$ with respect to $J$: $(L_{\mathbb R})^{\mathbb C} := (L_{\mathbb R}^2,J)$. Similarly, we have complexification $f^{\mathbb C}$ of $f \in End_{\mathbb R}(L_{\mathbb R})$ defined with respect to the canonical $J$: $f^{\mathbb C} := (f \oplus f)^J$.
- 2.1. For every $f \in End_{\mathbb R}(L_{\mathbb R})$, $f \oplus f$ commutes with $J$ i.e. $f^{\mathbb C} := (f \oplus f)^J$ is $\mathbb C$-linear.
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Let $i^{\sharp}$ be the unique almost complex structure on $L_{\mathbb R}$ such that $L=(L_{\mathbb R},i^{\sharp})$.
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Let $\hat i := i^{\sharp} \oplus i^{\sharp}$ such that $(\hat i)^J = (i^{\sharp})^{\mathbb C}$. Then $\hat i$ is another almost complex structure on $L_{\mathbb R}^2$.
- 4.1. By (2.1), $\hat i$ and $J$ commute, i.e. both $(\hat i)^J$ and $J^{\hat i}$ are $\mathbb C$-linear. (See here for related question.)
Part I of my understanding:
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The authors compute the eigenvalues of $J^{\hat i}$ and not $(\hat i)^J$. Then, they compute the corresponding eigenspaces.
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We know 'eigenspaces are subspaces', so if someone were to ask
Why do the authors have to explain why $L^{1,0}$ and $L^{0,1}$ are $\mathbb C$-subspaces of $(L_{\mathbb R})^{\mathbb C}$?,
then the answer would be:
We know $L^{1,0}$ and $L^{0,1}$ are $\mathbb C$-subspaces of $(L_{\mathbb R}^2,\hat i)$ by "eigenspaces are subspaces", but we want to also show that $L^{1,0}$ and $L^{0,1}$ are $\mathbb C$-subspaces of $(L_{\mathbb R}^2,J) = (L_{\mathbb R})^{\mathbb C}$.
Part II of my understanding: It is unnecessary for the authors to compute the eigenvalues of $J^{\hat i}$ and then show that the eigenspaces are $\mathbb C$-subspaces of $(L_{\mathbb R}^2,J) = (L_{\mathbb R})^{\mathbb C}$.
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The eigenvalues of $J^{\hat i}$ are the same as the eigenvalues of $(\hat i)^J$. The corresponding eigenspaces also have the same underlying sets. (See here for related question.)
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By (7), the authors could have directly computed eigenvalues $(\hat i)^J$ and corresponding eigenspaces. Then, there is no need to explain why said eigenspaces would be $\mathbb C$-subspaces of $(L_{\mathbb R}^2,J) = (L_{\mathbb R})^{\mathbb C}$.
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8.1. Note: An $\mathbb R$-vector space $A$ has an almost complex structure $H$ if and only if $A=B_{\mathbb R}$ for some non-unique $\mathbb C$-vector space $B$, such as $B=(A,H)$.
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8.2. By (8.1), I believe (8) is precisely what Huybrechts (Chapter 1.2) does except Huybrechts uses arbitrary almost complex structure '$I$' on '$V$' instead of specifically what would be $i^{\sharp}(v): = iv$ on $V$, viewed as the realification of some $\mathbb C$-vector space $L$, i.e. viewed as $V = L_{\mathbb R}$. Furthermore, the 'i', '$I$' and '$I^{\mathbb C}$' of Huybrechts correspond, respectively, to the $J$, $i^{\sharp}$ and $(i^{\sharp})^{\mathbb C}$ of Suetin, Kostrikin and Mainin.
Best Answer
Just so this has an answer:
Yes, it is correct.