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I am starting reading about Hodge theory and while reading the definition of abstract Hodge structure a very basic question came to my mind... What is the definition of the conjugate of a subspace of a complex vector space?

I found online the definition of conjugate of an entire vector space, but when we do the Hodge decompostition we want $V_{\mathbb{C}}=\oplus_{p,q \in \mathbb{Z}}V^{p,q}$ with the requirement $\overline{V^{q,p}}=V^{p,q}$. So I want to take a conjugate inside the space $V_{\mathbb{C}}$. I can see it if my space is of differential forms, but what is the definition for an abstract one?

Also, if I take the tensor product of two vector spaces, is this conjugation induced componentwise?

I know the question is pretty basic, but I couldn't find any reference for such definitions.

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Let $V$ be a complex vector space. A complex conjugation on $V$ is an antilinear map $\sigma : V \to V$ such that $\sigma\circ\sigma = \operatorname{id}_V$. Not every complex vector space comes with a complex conjugation, in general it is an extra piece of data.

Complex Euclidean space $\mathbb{C}^n$ has a complex conjugation given by $\sigma(z_1, \dots, z_n) = (\overline{z_1}, \dots, \overline{z_n})$. As any finite dimensional complex vector space is non-canonically isomorphic to $\mathbb{C}^n$, we can always obtain a complex conjugation $\sigma_V$ on $V$ by choosing an isomorphism $\phi : V \to \mathbb{C}^n$ and setting $\sigma_V = \phi^{-1}\circ\sigma\circ\phi$.

Note that $\mathbb{C}^n$ is not the only complex vector space which carries a natural complex conjugation. If $W$ is a real vector space, the complex vector space $V = W\otimes_{\mathbb{R}}\mathbb{C}$ is called the complexification of $W$ and it carries a natural complex conjugation given by $\sigma(w\otimes z) = w\otimes\overline{z}$. This generalises the previous example as $\mathbb{C}^n = \mathbb{R}^n\otimes_{\mathbb{R}}\mathbb{C}$; you can check that the complex conjugation given by the complexification is the same as the one given above. Once you choose a complex conjugation on $V$, which as we showed above is always possible, you can write $V$ as the complexification of a real vector space $W$ (the set of vectors fixed by $\sigma$).

If $V$ has a complex conjugation $\sigma$, then given any subspace $W$, the conjugate subspace, often denoted $\overline{W}$, is $\sigma(W)$; you should check that this is indeed a subspace.

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  • $\begingroup$ Very interesting. Would you be so kind as to recommend a text book to look it up? The ones I have found do not dwell too much into complex vector spaces, and when they do they overlook conjugations. $\endgroup$ – Rogelio Molina May 17 '15 at 0:18
  • $\begingroup$ @RogelioMolina: I don't know of a book that discusses this. The notes by Keith Conrad mentioned above seem quite thorough. $\endgroup$ – Michael Albanese May 17 '15 at 3:35

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