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:
Definitions, Assumptions, Notations
Let $V$ be $mathbb R$-vector space, possibly infinite-dimensional.
Complexification of space definition: Its complexification can be defined as $V^{mathbb C} := (V^2,J)$ where $J$ is the almost complex structure $J: V^2 to V^2, J(v,w):=(-w,v)$ which corresponds to the complex structure $s_{(J,V^2)}: mathbb C times V^2 to V^2,$$ s_{(J,V^2)}(a+bi,(v,w))$$:=s_{V^2}(a,(v,w))+s_{V^2}(b,J(v,w))$$=a(v,w)+bJ(v,w)$ where $s_{V^2}$ is the real scalar multiplication on $V^2$ extended to $s_{(J,V^2)}$. In particular, $i(v,w)=(-w,v)$.
Note on Complexification of space definition: The above definition however depends on $J$, so to denote this dependence, we may write $V^{(mathbb C,J)}=V^{mathbb C}$. We could have another definition replacing $J$ with any other almost complex structure $K$ which necessarily relates to $J$ by $K = S circ J circ S^{-1}$ for some $S in Aut_{mathbb R}(V^2)$. For example with $K = – J$ (I think $S$ would be $S(v,w):=(v,-w)$, which is $mathbb C$-antilinear with respect to $J$, and even to $K=-J$ I think), we get $i(v,w)=(w,-v)$.
Complexification of map definition: Based on Conrad, Bell, Suetin, Kostrikin and Mainin (12.10-11 of Part I) and Roman (Chapter 2), it looks like we can define the complexification (with respect to $J$) $f^{mathbb C}: V^{mathbb C} to V^{mathbb C}$ of $f: V to V$, $f in End_{mathbb R}V$ as any of the following equivalent, I think, ways (Note: we could actually have different vector spaces such that $f: V to U$, but I’ll just talk about the case where $V=U$)
Definition 1. $f^{mathbb C}(v,w):=(f(v),f(w))$
- I think ‘$mathbb C$-linear (with respect to $J$)’ isn’t part of this definition but is deduced anyway.
Definition 2. $f^{mathbb C}$ the unique $mathbb C$-linear (with respect to $J$) map such that $f^{mathbb C} circ cpx = cpx circ f$, where $cpx: V to V^{mathbb C}$ is the complexification map, as Roman (Chapter 1) calls it, or the standard embedding, as Conrad calls it. (Note: I think $cpx$ doesn’t depend on $J$.)
Definition 3. $f^{mathbb C}$ the unique $mathbb C$-linear (with respect to $J$) map such that $(f^{mathbb C})_{mathbb R} = f oplus f$
Definition 4. $f^{mathbb C} := (f oplus f)^J$ and again ‘$mathbb C$-linear (with respect to $J$)’ isn’t part of this definition but is deduced anyway. Here, the notation $(cdot)^I$ is:
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Complex structure on map definition: The operator ‘$(cdot)^I$‘ is supposed to be something like an inverse of the realification functor $(cdot)_{mathbb R}$ (see Jordan Bell and Suetin, Kostrikin and Mainin). If $(cdot)^I$ is some kind of functor, then $W^I := (W,I)$.
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I couldn’t find any book that uses this kind of notation, but the point of this ‘$g^I$‘ is mainly to be specific and allow shortcuts. Example: The statement ‘$g$ is $mathbb C$-linear with respect to $I$‘ becomes just ‘$g^I$ is $mathbb C$-linear’. Another example: For any almost complex structure $K$ on $W$, $K^K$ is $mathbb C$-linear, but $I^K$ and $K^I$ are not necessarily $mathbb C$-linear. However, with $-I$ as another almost complex structure on $W$, I think $I^{-I}$and ${-I}^{I}$ are $mathbb C$-linear.
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Proposition: $g^I$ is $mathbb C$-linear if and only if $g$ is $mathbb R$-linear and $g$ ‘commutes with scalar multiplication by i (with respect to $I$)’, meaning $g circ I = I circ g$.
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We can also extend to defining maps like $g^{(I,H)}: (W,I) to (U,H)$ and saying $g^{(I,H)}$ is $mathbb C$-linear if and only if $g$ is $mathbb R$-linear and $g circ I = H circ g$. In this notation and for the case of $W=U$, $g^{(I,I)}=g^I$.
Regardless of the definition, we end up with the formula given in Definition 1 (Even if the definitions aren’t equivalent, whichever definitions are correct, I think will give this formula in Definition 1).
Note on Complexification of map definition: The above definition/s however depends on $J$, so to denote this dependence, we may write $f^{(mathbb C,J)}=f^{mathbb C}$.
Questions:
Question 1: What is the formula for $f^{(mathbb C,K)}$ for any almost complex structure $K$ on $V^2$, assuming it exists, whether uniquely or not?
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Note: I actually didn’t think $f^{(mathbb C,K)}$ wouldn’t be unique or even exist until mid way through typing this (so I added 2 more questions below), so there might be kind of a definition issue here, but I guess it’s ok to define $f^{(mathbb C,K)}$ as any $mathbb C$-linear (with respect to $K$) map such that $f^{(mathbb C,K)} circ cpx = cpx circ f$
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Example: For $K=-J$, I think we get still $f^{(mathbb C,-J)}(v,w)=(f(v),f(w))$ (I derived this in a similar way that Conrad derived the formula for $K=J$).
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Example: Suppose $V$ in turn has an almost complex structure $k$. Then $k oplus k$ is an almost complex structure on $V^2$. For $K=k oplus k$, I don’t know how to get the formula for $f^{(mathbb C,k oplus k)}(v,w)$, similar to the cases of $K= pm J$. Maybe it doesn’t exist.
Question 2: Does $f^{(mathbb C,K)}$ always exist even if not uniquely?
Question 3: Whenever $f^{(mathbb C,K)}$ exists, is $f^{(mathbb C,K)}$ unique?
Note: This question might be answered by the answer, that I’m still analysing, to another question I posted.
More thoughts based on these:
It appears that:
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complexification relies not only on an almost complex structure $K$ on $V^2$ but also on a choice of subspace $A$ of $V^2$, where $A$ is not $V^2$ or $0$. This $A$ is what we use to identify $V$ as an embedded $mathbb R$-subspace of $V^2$
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For any subspace $A$ of $V^2$, except $V^2$ and $0$, and for any almost complex structure $K$ on $V^2$, there exists a unique involutive $mathbb R$-linear map $sigma_{A,K}$, on $V^2$, such that $sigma_{A,K}$ anti-commutes with $K$ and the set of fixed points of $sigma_{A,K}$ is equal to $A$.
- 2.1. For example, $sigma_{V times 0,J} = chi$, where $chi(v,w):=(v,-w)$
- Therefore, I should ask about $f^{(mathbb C,K,A)}$, not $f^{(mathbb C,K)}$.
