Conjugate-linear Hodge star
= Conjugate-linear Hodge star
{title2=$*:\mathcal A^{p,q}\to\mathcal A^{n-p,n-q}$}
For a <Hermitian manifold> with the pointwise <Hermitian inner product> linear in the first variable, define the conjugate-linear star by $\alpha\wedge *\beta=\langle\alpha,\beta\rangle\,\omega^n/n!$. It is the usual real <Hodge star operator> extended complex-linearly and then composed with complex conjugation. It has the displayed bidegree and satisfies $*^2=(-1)^{p+q}$ on $(p,q)$-forms.