= Hodge star on top holomorphic forms
{c}
{title2=$*\eta=(-1)^{n(n+1)/2}i^n\eta$}
For the complex-linear <Hodge star operator> and complex orientation on a <Hermitian manifold> of complex dimension $n$, every $(n,0)$-form has the displayed eigenvalue. In an adapted orthonormal real coframe, each factor $dx_j+i\,dy_j$ has two-dimensional star eigenvalue $-i$. Regrouping the $n$ factors introduces $(-1)^{n(n-1)/2}$, giving $(-1)^{n(n+1)/2}i^n$. This is a pointwise metric identity; the metric need not be <Kähler>.
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