Questions on the complexification of a map

abstract-algebraalmost-complexcomplex-analysiscomplex-geometrylinear algebra

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 $V$ be $\mathbb R$-vector space, which may be infinite-dimensional and which may or may not have an almost complex structure. Then $V^2$ necessarily has an almost complex structure such that we can define complexification of $V$ as $V^{\mathbb C} := (V^2,J)$, the unique $\mathbb C$-vector space corresponding to the canonical almost complex structure $J: V^2 \to V^2$ is $J(v,w):=(-w,v)$.

Questions:

Question 1. Is the map that yields a complexification unique?

  • Let $H: V^{\mathbb C} \to V^{\mathbb C}$ be the complexification of a map $h \in End_{\mathbb R}(V)$, i.e. $H=h^{\mathbb C}=(h \oplus h)^J$, following the notation here (specifically the bullet below Definition 4).

  • Is $h$ unique, i.e. if $H=h^{\mathbb C}=(h \oplus h)^J=g^{\mathbb C}=(g \oplus g)^J$ for some $g \in End_{\mathbb R}(V)$, then $h=g$?

Question 2. If an almost complex structure on $V^2$ is the complexification of a map on $V$, then is that map an almost complex structure on $V$?

  • If $V$ has an almost complex structure $h$, then $h \oplus h$ is an almost complex structure on $V^2$.

  • If $V$ does not necessarily have an almost complex structure but $V^{\mathbb C}$ has a map $F=f^{\mathbb C}$ for some $f \in End_{\mathbb R}(V)$ and $F_{\mathbb R}=f \oplus f$ is an almost complex structure on $V^2$, then is $f$ actually an almost complex structure on $V$?

Question 3. For Suetin, Kostrikin and Mainin, 12.2 of Part I, 12.5-7 of Part I and 12.10-11 of Part I:

  • In 12.10-11 of Part I, why don't we have $(L^{\mathbb C})_{\mathbb R}$ and $(f^{\mathbb C})_{\mathbb R}$ as 'literally' ('literally' is meant as in here) the same as, respectively, $L^2 = L \bigoplus L$ and $f \oplus f$?

Question 4. Is $k \oplus h$ an almost complex structure on $V^2$ if and only if $k$ and $h$ are almost complex structures on $V$?

Best Answer

Question 1: Yes because $(h \oplus h)^J=(g \oplus g)^J$ iff $h \oplus h=g \oplus g$ iff $h=g$.

Question 2: Yes for same reason Question 1 is yes: $F_{\mathbb R} \circ F_{\mathbb R} = -id_{V^2} = -(id_V \oplus id_V) = (-id_V) \oplus (-id_V)$ and $F_{\mathbb R} \circ F_{\mathbb R} = (f \oplus f) \circ (f \oplus f) = (f \circ f) \oplus (f \circ f)$

Question 3: I just assume literal.

Question 4: Yes because $(k \oplus h) \circ (k \oplus h) = (k \circ k) \oplus (h \circ h)$ and $-id_{V^2} = -id_{V} \oplus -id_{V}$

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