## Does the Hodge star operator respect complex structure?

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The Hodge star operator $\ast$ acts on the differential forms of a differential manifold sending $\Omega^{k}$ to $\Omega^{N-k}$. If the manifold is complex, then for $p+q=k$, does $\ast$ map $\Omega^{p,q}$ into some $\Omega^{a,b}$, where $a+b=N-k$.

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The last question is ambiguous, but I think the 2nd sentence is also a question, so you should put a question mark after it. You can find the answer in many places, e.g. Griffiths-Harris page 66. – Donu Arapura Nov 21 2010 at 19:14
So yes in the sense you mean. – Donu Arapura Nov 21 2010 at 19:18
The last question is just a rewording of the previous line. Since it is non-essential and apparently confusing I'll delete it. – Abtan Massini Nov 21 2010 at 19:24

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Since $∗$ is a real operator, to be more precise, you should say that after complexifying the space of forms, and extending $∗$ to be complex linear, then indeed $∗$ maps $\Omega^{p,q}$ to $\Omega^{N-q,N-p}$. Frequently, it is preferable to use $\bar *$, which is the composition of $∗$ with complex conjugation, to map $\Omega{p,q}$ to $\Omega{N−p,N−q}$. This way, we have $\alpha \wedge \bar * \beta = g(\alpha, \beta) vol$, where $g$ is the Hermitian metric. – Spiro Karigiannis Nov 21 2010 at 19:42

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Our comments seemed to have crossed. Yes, p 82 not 66, and yes. (Be warned that some authors use a different convention, where $N-p,N-q$ get switched. It's a question of whether $*$ is linear or antilinear.) – Donu Arapura Nov 21 2010 at 19:42
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## 1 Answer

As Abtan requested, I'm converting my comments to an answer:

Suppose that $X$ is an $N$ (complex) dimensional complex manifold endowed with a Hermitean metric, or equivalently a Riemannian metric g satisfying $g(JX,JY)=g(X,Y)$, where $J$ is the complex structure. Let `$*$` denote the `$\mathbb{C}$`-antilinear extension of the Hodge star operator to complex valued forms (some people -- including me -- prefer to write this as `$\overline{*}$` as Spiro points out in the comments). Then as one finds on page 82 of Griffiths and Harris, `$$*\Omega^{pq}\subset \Omega^{N-q,N-p}$$` where I'm following the notation in the question and writing $\Omega^{pq}$ for the space of $C^\infty$ forms of type $(p,q)$.

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@Donu: there was a display problem with a math expression. I hope you don't mind that I fixed it.  – Willie Wong Nov 21 2010 at 20:03
Willie, I noticed that too. We might have been editing at the same time (?). – Donu Arapura Nov 21 2010 at 20:05
Oh and thanks. Looks better. – Donu Arapura Nov 21 2010 at 20:06
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