I was reading harmonic theory from Griffiths Harris Principles of Algebraic geometry. I have difficulty in verifying the definition of the star operator.
We have a Hermitian metric on the holomorphic tangent bundle and $\{\phi_{1}, \dots, \phi_{n}\}$ is a local unitary coframe. Then
$$ \omega = \frac{\sqrt{-1}}{2} \sum_{j}\phi_{j} \wedge \bar{\phi_{j}}. $$ Then $$ \Phi = \frac{\omega^{n}}{n!} = C_{n}\phi_{1}\wedge \dots \wedge \phi_{n} \wedge\bar{\phi_{1}} \wedge \dots \wedge \bar{\phi_{n}} $$ where $C_{n} = (-1)^{n(n-1)/2}(\sqrt{-1}/2)^{n}$.
Now in $T_{z}^{*(p,q)}M$ , $\{\phi_{I}\wedge \bar{\phi_{J}}\}$ are orthogonal with $||\phi_{I}\wedge \bar{\phi_{J}}||^{2} = 2^{p+q}$. This gives an Hermitian inner product on $T^{*(p,q)}_{z}M$.
The book then defines the star operator $* : A^{p,q}(M) \to A^{n-p, n- q}(M)$ by requiring $$ (\psi(z),\eta(z)) \Phi(z) = \psi(z) \wedge * \eta(z). $$ It then claims that for $$ \eta = \sum_{I,J} \eta_{I\bar{J}} \phi_{I}\wedge \bar{\phi}_{J} $$ we have $$ *\eta = 2^{p+q-n} \sum_{I,J} \varepsilon_{I,J}\bar{\eta}_{I\bar{J}}\phi_{I_{0}}\wedge \bar{\phi_{J_{0}}}. $$ Here $I_{0} = (1,2,\dots,n) - I$ and $\varepsilon_{I,J}$ is the sign of the permutation $$ (1,2,\dots,n, 1',2', \dots, n') \to (i_{1},\dots,i_{p}, j_{1},\dots, j_{q}, i_{1}^{0},\dots,i_{n-p}^{0}, j_{1}^{0}, \dots,j_{n-q}^{0}). $$
I think there is some error here. For example, if I assume that $\eta = \phi_{1}$, then according to the formula we should have $$ *\eta = 2^{1-n}\phi_{2,\dots,n} \wedge \bar{\phi}_{1,\dots,n}. $$
But this $*\eta$ doesn't satisfy the required equation that $(\psi(z),\eta(z))\Phi(z) = \phi(z)\wedge *\eta(z)$.
We have for $\psi(z) = \psi_{i}(z)\phi_{i}$, $$ \psi(z)\wedge *\eta(z) = 2^{1-n}\psi_{1}(z)\phi_{1}\wedge \dots \wedge \phi_{n} \wedge \bar{\phi}_{1} \wedge \dots \wedge \bar{\phi}_{n} $$ $$ = 2^{1-n}\psi_{1}(z)\Phi(z)/C_{n}. $$ While $$ (\psi(z),\eta(z))\Phi(z) = 2\psi_{1}(z)\Phi(z). $$ So I'm missing a factor of $(-1)^{n(n-1)/2}(\sqrt{-1})^{n}$.
Is there a mistake in the book or have I done some computation error?