1. 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?

2. If yes to Question 1, then is $I$ necessarily $I=k \oplus h$ for some almost complex structures $k$ and $h$?

3. 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?

1. 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?

2. Whenever we have such an $I$, is $I$ necessarily $I=k \oplus h$ for some almost complex structures $k$ and $h$?

3. 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?

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 $\hat i: V_{\mathbb R}^2 \to V_{\mathbb R}^2$, $\hat i := i^{\sharp} \oplus i^{\sharp}$.

1. $\chi^J$ is a conjugation, on $(V_{\mathbb R})^{\mathbb C}$, called the standard conjugation on $(V_{\mathbb R})^{\mathbb C}$.

2. $\hat i$ is an almost complex structure on $V_{\mathbb R}^2$.

3. While $\chi^J$ and $\chi^{-J}$ are $\mathbb C$-anti-linear, we have that $\chi^{\hat i}$ is $\mathbb C$-linear.

4. $k$ and $h$ are almost complex structures on $V_{\mathbb R}$ if and only if $k \oplus h$ is an almost complex structure on $V_{\mathbb R}^2$

5. 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 $k=h=i^{\sharp}$.

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 $V=(V_{\mathbb R},i^{\sharp})$.

1. $\chi^J$ is a conjugation, on $(V_{\mathbb R})^{\mathbb C}$, called the standard conjugation on $(V_{\mathbb R})^{\mathbb C}$.

2. 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 $V_{\mathbb R}^2$.

3. While $\chi^J$ and $\chi^{-J}$ are $\mathbb C$-anti-linear, we have that $\chi^{\hat i}$ is $\mathbb C$-linear.

4. $k$ and $h$ are almost complex structures on $V_{\mathbb R}$ if and only if $k \oplus h$ is an almost complex structure on $V_{\mathbb R}^2$

5. 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 $k=h=i^{\sharp}$.

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 $i^{\sharp}: V_{\mathbb R} \to V_{\mathbb R}$ denote the almost complex structure on $V_{\mathbb R}$, given by $i^{\sharp}(v) := iv$. Let $\hat i: V_{\mathbb R}^2 \to V_{\mathbb R}^2$, $\hat i(v,w) := i^{\sharp} \oplus i^{\sharp}$.

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 $\hat i: V_{\mathbb R}^2 \to V_{\mathbb R}^2$, $\hat i := i^{\sharp} \oplus i^{\sharp}$.