Let $X$ be a connected compact Kähler complex manifold of complex dimension $n\ge 1$ with holomorphically trivial canonical bundle $K_X$, and let $\Omega\in H^0(X,K_X)$ be a nowhere-vanishing holomorphic $n$-form. Fix a Kähler class $\kappa\in H^2(X,\mathbb R)$, let $\omega$ be the unique Ricci-flat Kähler form in $\kappa$, and let $(S,0)$ be a smooth local polarized deformation space of the pair $(X,\kappa)$, represented by a smooth proper holomorphic Kähler family $\pi:\mathcal X\to S$ with central fibre $X_0=X$ and with a locally constant polarization class whose restriction to $X$ is $\kappa$. Assume the Kodaira-Spencer map identifies $T_0S$ with the corresponding subspace of $H^1(X,T_X)$.

For $v,w\in T_0S$, let $\varphi,\psi\in A^{0,1}(X,T_X)$ be the $\omega$-[harmonic representatives](/theorems/2747) of the corresponding Kodaira-Spencer classes. Let $[X]$ denote the oriented fundamental class of the underlying real manifold of $X$, and let $\mathcal I_X:A^{n,n}(X)\to\mathbb C$ denote integration of top-degree smooth complex forms over the oriented real manifold $X$, equivalently evaluation of the de Rham class on $[X]$. Define the Hodge norm of the period line generated by $\Omega$ by

\begin{align*} \|\Omega\|^2:=i^{n^2}\mathcal I_X(\Omega\wedge\overline{\Omega}). \end{align*}

Assume the Weil-Petersson Hermitian form is defined as the positive curvature Hermitian form of the Hodge metric on the Hodge line $F^n=H^{n,0}$, with the primitive Hodge-Riemann convention on polarized primitive classes in $H^{n-1,1}(X)$ given by

\begin{align*} (\alpha,\beta)_{\mathrm{HR}}:=(-1)^{n(n-1)/2}i^{n-2}\mathcal I_X(\alpha\wedge\overline{\beta}). \end{align*}

Here primitive means annihilated by cup product with $\kappa$, equivalently by wedging harmonic representatives with $\omega$. Then the polarized infinitesimal period classes $[\varphi\lrcorner\Omega]$ and $[\psi\lrcorner\Omega]$ are primitive, and

\begin{align*} g^{\mathrm{WP}}_0(v,w)=\frac{(-1)^{n(n-1)/2}i^{n-2}\mathcal I_X((\varphi\lrcorner\Omega)\wedge \overline{(\psi\lrcorner\Omega)})}{i^{n^2}\mathcal I_X(\Omega\wedge\overline{\Omega})}. \end{align*}

Equivalently,

\begin{align*} g^{\mathrm{WP}}_0(v,w)=\frac{([\varphi\lrcorner\Omega],[\psi\lrcorner\Omega])_{\mathrm{HR}}}{\|\Omega\|^2}. \end{align*}