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.


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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.