# Complexification of realification: Why compute eigenvalues? [closed]

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 is one:

Question: Below, I describe what I understand is happening in Suetin, Kostrikin and Mainin (12.13 of Part I), where the authors prove for any $$\mathbb C$$-vector space $$L$$, $$L_{\mathbb R}^{\mathbb C} \cong L \bigoplus \overline L$$ (I also asked about this isomorphism here.) Also, I base my understanding on Daniel Huybrechts, Complex Geometry An Introduction (Chapter 1.2). Is my understanding, given in the two parts as follows, correct?

Part 0. Assumptions, definitions, notations:

1. Complex structure on map definition : See here.

2. On $$L_{\mathbb R}^2$$, we have almost complex structure $$J(l,m):=(-m,l)$$. $$J$$ is canonical in the sense that we define complexification $$(L_{\mathbb R})^{\mathbb C}$$ of $$L_{\mathbb R}$$ with respect to $$J$$: $$(L_{\mathbb R})^{\mathbb C} := (L_{\mathbb R}^2,J)$$. Similarly, we have complexification $$f^{\mathbb C}$$ of $$f \in End_{\mathbb R}(L_{\mathbb R})$$ defined with respect to the canonical $$J$$: $$f^{\mathbb C} := (f \oplus f)^J$$.

• 2.1. For every $$f \in End_{\mathbb R}(L_{\mathbb R})$$, $$f \oplus f$$ commutes with $$J$$ i.e. $$f^{\mathbb C} := (f \oplus f)^J$$ is $$\mathbb C$$-linear.
1. Let $$i^{\sharp}$$ be the unique almost complex structure on $$L_{\mathbb R}$$ such that $$L=(L_{\mathbb R},i^{\sharp})$$.

2. Let $$\hat i := i^{\sharp} \oplus i^{\sharp}$$ such that $$(\hat i)^J = (i^{\sharp})^{\mathbb C}$$. Then $$\hat i$$ is another almost complex structure on $$L_{\mathbb R}^2$$.

• 4.1. By (2.1), $$\hat i$$ and $$J$$ commute, i.e. both $$(\hat i)^J$$ and $$J^{\hat i}$$ are $$\mathbb C$$-linear. (See here for related question.)

Part I of my understanding:

1. The authors compute the eigenvalues of $$J^{\hat i}$$ and not $$(\hat i)^J$$. Then, they compute the corresponding eigenspaces.

2. We know 'eigenspaces are subspaces', so if someone were to ask

Why do the authors have to explain why $$L^{1,0}$$ and $$L^{0,1}$$ are $$\mathbb C$$-subspaces of $$(L_{\mathbb R})^{\mathbb C}$$?,

We know $$L^{1,0}$$ and $$L^{0,1}$$ are $$\mathbb C$$-subspaces of $$(L_{\mathbb R}^2,\hat i)$$ by "eigenspaces are subspaces", but we want to also show that $$L^{1,0}$$ and $$L^{0,1}$$ are $$\mathbb C$$-subspaces of $$(L_{\mathbb R}^2,J) = (L_{\mathbb R})^{\mathbb C}$$.

Part II of my understanding: It is unnecessary for the authors to compute the eigenvalues of $$J^{\hat i}$$ and then show that the eigenspaces are $$\mathbb C$$-subspaces of $$(L_{\mathbb R}^2,J) = (L_{\mathbb R})^{\mathbb C}$$.

1. The eigenvalues of $$J^{\hat i}$$ are the same as the eigenvalues of $$(\hat i)^J$$. The corresponding eigenspaces also have the same underlying sets. (See here for related question.)

2. By (7), the authors could have directly computed eigenvalues $$(\hat i)^J$$ and corresponding eigenspaces. Then, there is no need to explain why said eigenspaces would be $$\mathbb C$$-subspaces of $$(L_{\mathbb R}^2,J) = (L_{\mathbb R})^{\mathbb C}$$.

• 8.1. Note: An $$\mathbb R$$-vector space $$A$$ has an almost complex structure $$H$$ if and only if $$A=B_{\mathbb R}$$ for some non-unique $$\mathbb C$$-vector space $$B$$, such as $$B=(A,H)$$.

• 8.2. By (8.1), I believe (8) is precisely what Huybrechts (Chapter 1.2) does except Huybrechts uses arbitrary almost complex structure '$$I$$' on '$$V$$' instead of specifically what would be $$i^{\sharp}(v): = iv$$ on $$V$$, viewed as the realification of some $$\mathbb C$$-vector space $$L$$, i.e. viewed as $$V = L_{\mathbb R}$$. Furthermore, the 'i', '$$I$$' and '$$I^{\mathbb C}$$' of Huybrechts correspond, respectively, to the $$J$$, $$i^{\sharp}$$ and $$(i^{\sharp})^{\mathbb C}$$ of Suetin, Kostrikin and Mainin.

• Btw, I think the term "complexification", in the meaning of "making it complex", or "do $-\otimes_{\mathbb{R}}\mathbb{C}$", was very well found (who invented it? Arnol'd?) On the opposite, the later neologism "realification", to denote "forgetting the complex structure", seems less satisfactory. It appears as a compound of "facio", but nothing is really "done", and no structure is added. The old "decomplexify" seems more accurate to me. Nov 13, 2020 at 14:47
• @PietroMajer Realification is the $\mathbb C$ to $\mathbb R^2$ and not $\mathbb C$ to $\mathbb R$ (or $\mathbb R+0i$, or $0+\mathbb Ri$) right? Nov 13, 2020 at 15:42
• Yes, I think so Nov 13, 2020 at 15:53
• @PietroMajer Do you wanna post 'yes it is correct' and then i offer the bounty to you? lol Nov 13, 2020 at 16:12