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: Let $W = (W,\text{Add}_W: W^2 \to W,s_W: \mathbb R \times W \to W)$ be an $\mathbb R$-vector space, which may be infinite-dimensional. Suppose $W$ has an almost complex structure $H \in Aut_{\mathbb R}(W)$. $H$ uniquely corresponds to the $\mathbb C$-vector space $(W,H)$, where scalar multiplication is given by the complex structure $s_W^{H}: \mathbb C \times W \to W$, $s_W^{H}(a+ib,v) := s_W(a,v) + s_W(b,H(v))$. Note that $s_W^{H}$ agrees with the original real scalar multiplication $s_W$.
What I know:
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A1. For $W$ finite-dimensional and for any other almost complex structure $J \in Aut_{\mathbb R}(W)$, we have that $(W,H)$ and $(W,J)$ are $\mathbb C$-isomorphic but not necessarily by the identity map $id_W$ on $W$.
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A2. For $W$ finite-dimensional and for any other almost complex structure $J \in Aut_{\mathbb R}(W)$, we have that $H$ and $J$ are similar, i.e. $H \circ S = S \circ J$, for some $S \in Aut_{\mathbb R}(W)$
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A3. (A2) may be proven using (A1). See here and here. (I'm not sure if Moore (Section 9.1) uses (A1)).
Questions:
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Is (A1) true if $W$ were instead infinite-dimensional? (You may use axiom of choice.)
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Is (A2) true if $W$ were instead infinite-dimensional? (You may use axiom of choice.)
What I've tried:
- If yes to Question 1, then I think yes to Question 2 because I think we can still make the same argument as in the answers in the links in (A3). If no to Question 1, I think we don't necessarily have no to Question 2.
Related question: Different almost complex structures: $\mathbb C$-isomorphism for $(W,K)$ and $(W,H)$
Best Answer
Yes, and I don't know of any proof in the finite-dimensional case that does not also work in the infinite-dimensional case with trivial modifications. Given an almost complex structure $J$ on $W$, you can think of $W$ as a $\mathbb{C}$-vector space via $J$. Pick a basis $B$ for this $\mathbb{C}$-vector space. Then $B\cup J(B)$ is a basis for $W$ as an $\mathbb{R}$-vector space. So, $\dim_\mathbb{R} W=2\cdot \dim_\mathbb{C} W$. In particular, this means that $\dim_\mathbb{C} W$ is uniquely determined by $\dim_\mathbb{R} W$ (it is half the dimension when it is finite, or the same dimension when it is infinite). So any other almost complex structure $H$ makes $W$ a $\mathbb{C}$-vector space of the same dimension as $J$ does, so they are isomorphic.
Note that question 2 is equivalent to question 1. Indeed, $S:W\to W$ witnesses that $J$ and $H$ are similar iff $S$ is an isomorphism $(W,J)\to (W,H)$ of complex vector spaces (this is trivial from the definitions).