Given a finite dimensional (pseudo)-Riemannian manifold $(X,g)$, the Hodge star operator “completes” a $k$-differential form to the volume form of $(X,g)$.
Let $(X,g)$ be an oriented $n$-dimensional smooth manifold $X$ endowed with a (pseudo)-Riemannian metric $g$. For $0 \leq k \leq n$, write $\Omega^k(X)$ for the vector space of $k$-forms on $X$.
The metric $g$ naturally induces a nondegenerate symmetric bilinear form
If $X$ is compact then the integral of this against the volume form $vol_g$ exists. This is the Hodge inner product
The Hodge star operator is the unique linear function
defined by the identity
where $vol_g \in \Omega^n X$ is the volume form induced by $g$.
Therefore in terms of the Hodge operator the Hodge inner product reads
If $e_1,\dots,e_n$ is a local basis on $X$ and $e^1,\dots,e^n$ is the dual basis, so that $\alpha = \frac{1}{k!} \alpha_{i_1,\dots,i_k} e^{i_1} \wedge \cdots \wedge e^{i_k}$, then
where $\epsilon_{i_1,\dots,i_n}$ is the sign of the permutation $(1,2,\dots,n)\mapsto (i_1,i_2,\dots,i_n)$ and $det(g)$ is the determinant of $g$ in the local basis.
Let $(X,g)$ be a Riemannian manifold of dimension $n$ and let $\omega,\lambda \in \Omega^k(X)$. Then
$\star(\star\omega) = (-1)^{k(n+1)} \omega = (-1)^{k(n-k)} \omega$;
$\langle\star\omega , \star\lambda\rangle = \langle\omega | \lambda\rangle$;
$\star 1 = vol$.
On a Kähler manifold $\Sigma$ of dimension $dim_{\mathbb{C}}(\Sigma) = n$ the Hodge star operator acts on the Dolbeault complex as
(notice the exchange of the role of $p$ and $q$). See e.g. (Biquerd-Höring 08, p. 79). See also at Serre duality.
The metric $g$ is used in two places in the specification of the Hodge operator: in the inner product on forms and in the volume form. If $X$ is equipped only with a volume form (not necessarily coming from a metric), then the Hodge operator still takes $k$-forms to $(n-k)$-vector fields. If the manifold is not oriented, then the metric only gives a volume pseudoform, but the Hodge operator still takes $k$-forms to $(n-k)$-pseudoforms. Finally, if $X$ is equipped with only a volume pseudoform (which is equivalent to an absolutely continuous Radon measure on $X$), then the Hodge operator takes $k$-forms to $(n-k)$-pseudovector fields. (Of course, in every case, one might apply the operator to pseudoforms or multivector fields to begin with.)
Some useful basic formulas are listed in
Discussion in complex geometry includes