Natural complex-linear isomorphism between $ V_J $ and $V^- $

Question

Let $V$ be a real vector space with a linear complex structure $J$ (see Wikipedia).

Denote $V_J$ the complex vector space induced from $V$ by the complex structure $J$.

Also, define the complexification of $V$ as usual $$ V^{\mathbb{C}} = V \otimes_{\mathbb{R}} \mathbb{C} $$ and let $\tilde{J}$ be the complex structure induced from $V$ to $V^{\mathbb{C}}$ $$ \tilde{J}(v \otimes \lambda) = J(v) \otimes \lambda \qquad v \in V \quad \lambda \in \mathbb{C} . $$

$V^{\mathbb{C}}$ then decomposes as $$ V^{\mathbb{C}} = V^+ \oplus V^- $$ where $V^{\pm}$ are the $\pm i$ eigenspaces of the operator $\tilde{J}$.

$$ V^{\pm} = \{ v \otimes 1 \mp Jv \otimes i : v \in V \} $$

Wikipedia claims the following:

There is a natural complex linear isomorphism between $V_J$ and $V^+$, so these vector spaces can be considered the same, while $V^-$ may be regarded as the complex conjugate of $V_J$.

My questions are:

1. In what sense is the isomorphism natural? Does it mean the same thing as canonical, i.e. basis-independent?
2. Is it also possible to construct such a natural/canonical complex-linear isomorphism between $V^-$ and $V_J$? If not, why?

Answer 1

The isomorphisms are given by

$$ v \mapsto \frac{1}{2} (v - \tilde{i} Jv) \colon (V, J) \rightarrow (V^{+}, \tilde{i}|_{V^{+}}), \\ v \mapsto \frac{1}{2} (v + \tilde{i}Jv) \colon (V,J) \rightarrow (V^{-},\tilde{i}|_{V^{-}}). $$

Both isomorphisms are basis independent. The first is $\mathbb{C}$-linear while the second is $\mathbb{C}$-antilinear which naturally identifies $(V^{-}, \tilde{i}|_{V^{-}})$ with the conjugate complex vector space $(V,-J)$ in a $\mathbb{C}$-linear way. Here, $\tilde{i}$ is the complex structure on $V^{\mathbb{C}}$ induced via the complexification (that is, $\tilde{i} = \operatorname{id}_V \otimes i$ where $i$ is the natural complex structure on $\mathbb{C}$ just like $\tilde{J} = J \otimes \operatorname{id}_{\mathbb{C}}$). It is usually denoted just by $i$, if at all.

Regarding your second question, a natural $\mathbb{C}$-linear isomorphism between $(V,J)$ and $(V^{-}, \tilde{i}|_{V^{-}})$ would give you a natural $\mathbb{C}$-linear isomorphism between $(V,J)$ and $(V,-J)$. This isomorphism should extend to the level of complex vector bundles so you would get that any complex vector bundle $E$ is $\mathbb{C}$-linear isomorphic to the conjugate bundle $\overline{E}$. However, it is known to be false in general (for example, using Chern classes).