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 are some:
Assumptions and notations: Let $V$ be a $\mathbb C$-vector space. Let $V_{\mathbb R}$ be the realification of $V$. For any almost complex structure $I$ on $V_{\mathbb R}$, denote by $(V_{\mathbb R},I)$ as the unique $\mathbb C$-vector space whose complex structure is given $(a+bi) \cdot v := av + bI(v)$. Let $i^{\sharp}$ be the unique almost complex structure on $V_{\mathbb R}$ such that $V=(V_{\mathbb R},i^{\sharp})$.
Let $W$ be an $\mathbb R$-vector space. Let $W^{\mathbb C}$ denote the complexification of $W$ given by $W^{\mathbb C} := (W^2,J)$, where $J$ is the canonical almost complex structure on $W^2$ given by $J(v,w):=(-w,v)$. Let $\chi: W^2 \to W^2$, $\chi(v,w):=(v,-w)$
For any map $f: V_{\mathbb R} \to V_{\mathbb R}$ and for any almost complex structure $I$ on $V_{\mathbb R}$, denote by $f^I$ as the unique map $f^I: (V_{\mathbb R}, I) \to (V_{\mathbb R}, I)$ such that $(f^I)_{\mathbb R} = f$. With this notation, the conditions '$f$ is $\mathbb C$-linear with respect to $I$' and '$f$ is $\mathbb C$-anti-linear with respect to $I$' are shortened to, respectively, '$f^I$ is $\mathbb C$-linear' and '$f^I$ is $\mathbb C$-anti-linear'.
The complexification, under $J$, of any $g \in End_{\mathbb R}W$ is $g^{\mathbb C} := (g \oplus g)^J$, i.e. the unique $\mathbb C$-linear map on $W^{\mathbb C}$ such that $(g^{\mathbb C})_{\mathbb R} = g \oplus g$
Let $\sigma: V_{\mathbb R}^2 \to V_{\mathbb R}^2$, $\gamma: W^2 \to W^2$ and $\eta: V_{\mathbb R} \to V_{\mathbb R}$ be any maps such that $\sigma^J$, $\gamma^J$ and $\eta^{i^{\sharp}}$ are conjugations. (The $J$'s are of course different, but they have the same formula.)
Questions:
For $\sigma$, does there exist an almost complex structure $I$ on $V_{\mathbb R}^2$ such that $\sigma^I$ is $\mathbb C$-linear, and why/why not?
Whenever we have such an $I$, is $I$ necessarily $I=k \oplus h$ for some almost complex structures $k$ and $h$?
For $\gamma$, does there exist an almost complex structure $K$ on $W^2$ such that $\gamma^K$ is $\mathbb C$-linear, and why/why not?
- Note: I think the answer to Question 3 is no if the answer to Question 1 is no. However, I think Question 3 is answered affirmatively and with explanation if the answer to Question is 1 is yes and the answer to Questions 2 is no.
For $\eta$, does there exist an almost complex structure $H$ on $V_{\mathbb R}$ such that $\gamma^K$ is $\mathbb C$-linear, and why/why not?
- Note: I think the answer to Question 4 is no if the answer to Question 3 is no.
Observations that led to above questions:
$\chi^J$ is a conjugation, on $(V_{\mathbb R})^{\mathbb C}$, called the standard conjugation on $(V_{\mathbb R})^{\mathbb C}$.
Let $\hat i: V_{\mathbb R}^2 \to V_{\mathbb R}^2$, $\hat i := i^{\sharp} \oplus i^{\sharp}$. $\hat i$ is an almost complex structure on $V_{\mathbb R}^2$.
While $\chi^J$ and $\chi^{-J}$ are $\mathbb C$-anti-linear, we have that $\chi^{\hat i}$ is $\mathbb C$-linear.
$k$ and $h$ are almost complex structures on $V_{\mathbb R}$ if and only if $k \oplus h$ is an almost complex structure on $V_{\mathbb R}^2$
Actually, I think $\chi^{k \oplus h}$ is $\mathbb C$-linear, for any almost complex structures $k$ and $h$ on $V_{\mathbb R}$, not just $k=h=i^{\sharp}$.