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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:

Assumptions, definitions and notations: Let $V$ be an $\mathbb R$-vector space. Define $K \in Aut_{\mathbb R} (V^2)$ as anti-involutive if $K^2 = -id_{V^2}$. Observe that $K$ is anti-involutive on $V^2$ if and only if $K$ is an almost complex structure on $V^2$. Let $\Gamma(V^2)$ be the $\mathbb R$-subspaces of $V^2$ that are isomorphic to $V$. Let $AI(V^2)$ and $I(V^2)$ be, respectively, the anti-involutive and involutive maps on $V^2$. Let $A \in \Gamma(V^2)$ and $K \in AI(V^2)$.

Note: My question as follows is related to this question, which asks if there exists a unique $\sigma \in I(V^2)$ that both anti-commutes with $K$ and has $A$ as equal to the set of its fixed points.

Question: If there exists such a $\sigma \in I(V^2)$, then is $K(A)$ equal to the fixed points of $-\sigma$?

BCLC
  • 13,459

1 Answers1

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Yes.

Proof:

  1. Observe that $K(fixed(\eta)) \subseteq fixed(-\eta)$ for both $\eta=\pm \sigma$.

    • 1.1. For $\eta=\sigma$, $K(A) \subseteq fixed(-\sigma)$.

    • 1.2. For $\eta=-\sigma$, $K(fixed(-\sigma)) \subseteq A$.

  2. Apply $K$ to latter set equality to get $fixed(-\sigma) \subseteq K(A)$.

Note: I hopefully do not assume either that $fixed(\sigma) \cong fixed(-\sigma)$ or that $fixed(\sigma) \bigoplus fixed(-\sigma)=V^2$.

BCLC
  • 13,459