Complex Hodge star operator

The Hodge star extends complex-linearly to complex differential forms. With the Hermitian pointwise inner product, it is characterized byOn a Hermitian manifold it maps forms of type $(p,q)$ to forms of type $(n-q,n-p)$.

Use the complex-linear Hodge star operator on the direct sum of all form degrees. Its square on degree $r$ is $(-1{)}^{r(n-r)}$. In positive odd dimension this is always one; in positive even dimension its sign is positive on even-degree forms and negative on odd-degree forms. Complementary basis wedges provide both roots of each sign, so the displayed eigenvalues all occur. On a single degree the Hodge star operator is an endomorphism only in middle degree; on middle degree $m$ in dimension $2m$ its possible eigenvalues are $\pm 1$ for even $m$ and $\pm i$ for odd $m$.

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 $d{x}_j+id{y}_j$ 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.