[proofplan] Choose a local holomorphic frame $\Omega_s$ for the Hodge line $F^n=H^{n,0}$ near $0$ and write its Hodge norm as $h(s)=i^{n^2}\mathcal I_s(\Omega_s\wedge\overline{\Omega_s})$, where $\mathcal I_s$ denotes integration of top-degree forms over the oriented fibre $X_s$. The curvature of this Hermitian line is computed by differentiating $\log h$, and the logarithmic correction subtracts the component of the first variation lying along the line itself. [Griffiths transversality](/theorems/9129) and the infinitesimal period map identify the remaining horizontal component in the direction $v$ with the class represented by $\varphi\lrcorner\Omega$. Finally the primitive Hodge-Riemann convention converts the Hodge norm of this $(n-1,1)$ component into the displayed integral formula. [/proofplan]

[step:Choose a local Hodge frame and write the curvature formula for its norm]Let $\mathcal H^n:=R^n\pi_*\mathbb C\otimes_{\mathbb C}\mathcal O_U$ denote the flat cohomology bundle over a neighbourhood $U\subset S$ of $0$, equipped with its holomorphic structure and Gauss-Manin connection, and let $F^n\subset \mathcal H^n$ denote the Hodge line bundle with fibre $H^{n,0}(X_s)$. Since $X$ is connected and $K_X$ is holomorphically trivial, every holomorphic section of $K_X$ is a scalar multiple of the nowhere-vanishing section $\Omega$; hence $H^{n,0}(X)$ is the line generated by $\Omega$. Choose a local holomorphic section \begin{align*} \Omega_{\bullet}:U\to F^n \end{align*} of this line bundle such that $\Omega_0=\Omega$. For each $s\in U$, let $\mathcal I_s:A^{n,n}(X_s)\to\mathbb C$ denote integration of top-degree smooth complex forms over the oriented real manifold $X_s$, equivalently evaluation of the de Rham class on the oriented fundamental class $[X_s]$. Define the Hodge norm function \begin{align*} h:U\to \mathbb R_{>0} \end{align*} by \begin{align*} h(s):=i^{n^2}\mathcal I_s(\Omega_s\wedge\overline{\Omega_s}). \end{align*} Let $\partial_v$ denote differentiation at $0$ in the holomorphic tangent direction $v\in T_0S$, and let $\partial_{\bar w}$ denote differentiation at $0$ in the anti-holomorphic tangent direction $\bar w\in \overline{T_0S}$. By the sign convention in the statement, the Weil-Petersson Hermitian form is the positive curvature Hermitian form of the Hodge metric on $F^n$, hence in the local frame $\Omega_{\bullet}$ it is \begin{align*} g^{\mathrm{WP}}_0(v,w)=-\partial_v\partial_{\bar w}\log h\big|_0. \end{align*} The value is independent of the chosen local holomorphic frame because replacing $\Omega_{\bullet}$ by $f\Omega_{\bullet}$, where $f:U\to\mathbb C^\times$ is holomorphic, replaces $\log h$ by $\log h+\log |f|^2$, and $\partial_v\partial_{\bar w}\log |f|^2=0$.[/step]

[guided]The Hodge line is a Hermitian holomorphic line bundle. Because $X$ is connected and $K_X$ is holomorphically trivial, the quotient of any two nowhere-vanishing holomorphic sections of $K_X$ is a [holomorphic function](/page/Holomorphic%20Function) on compact connected $X$, hence is constant. Thus $H^{n,0}(X)$ is the one-dimensional line generated by $\Omega$. To compute the curvature of a Hermitian line bundle, one may choose any non-vanishing local holomorphic frame and differentiate the logarithm of the squared norm of that frame. Here the local frame is a holomorphic family of holomorphic volume forms \begin{align*} \Omega_{\bullet}:S\supset U\to F^n \end{align*} with value $\Omega_0=\Omega$ at the base point. For each $s\in U$, let $\mathcal I_s:A^{n,n}(X_s)\to\mathbb C$ denote integration of top-degree smooth complex forms over the oriented real manifold $X_s$, equivalently evaluation of the de Rham class on the oriented fundamental class $[X_s]$. The squared Hodge norm of this frame is the function \begin{align*} h:U\to \mathbb R_{>0} \end{align*} defined by \begin{align*} h(s):=i^{n^2}\mathcal I_s(\Omega_s\wedge\overline{\Omega_s}). \end{align*} The factor $i^{n^2}$ is part of the Hodge metric convention and makes the norm positive on the line $H^{n,0}$. For tangent vectors $v,w\in T_0S$, write $\partial_v$ for holomorphic differentiation at $0$ in the direction $v$, and $\partial_{\bar w}$ for anti-holomorphic differentiation at $0$ in the conjugate direction $\bar w$. With the sign convention fixed in the statement, the Weil-Petersson form is the positive curvature Hermitian form of the Hodge metric, so in the frame $\Omega_{\bullet}$ it is \begin{align*} g^{\mathrm{WP}}_0(v,w)=-\partial_v\partial_{\bar w}\log h\big|_0. \end{align*} This formula does not depend on the chosen frame. Indeed, if another local frame is $f\Omega_{\bullet}$ for a holomorphic map $f:U\to\mathbb C^\times$, then the squared norm changes from $h$ to $|f|^2h$. Therefore $\log h$ changes by $\log |f|^2$. Since $\log |f|^2$ is locally the sum of a holomorphic function and its complex conjugate, its mixed derivative $\partial_v\partial_{\bar w}$ is zero. Thus the curvature expression is intrinsic.[/guided]

[step:Decompose the first variation into its period-line part and horizontal part]Let $\nabla$ denote the Gauss-Manin connection on $\mathcal H^n$. The hypotheses give a smooth proper polarized Kähler family, so [Griffiths Transversality][citetheorem:9129] applies to the Hodge filtration on $\mathcal H^n$ and gives \begin{align*} \nabla_v\Omega_{\bullet}\big|_0\in F^{n-1}H^n(X,\mathbb C). \end{align*} Using the [Hodge decomposition](/theorems/2745) at $0$, define $a_v\in\mathbb C$ and $\eta_v\in H^{n-1,1}(X)$ by \begin{align*} \nabla_v\Omega_{\bullet}\big|_0=a_v\Omega+\eta_v. \end{align*} Define $a_w\in\mathbb C$ and $\eta_w\in H^{n-1,1}(X)$ by applying the same [Hodge decomposition](/theorems/3941) to the Gauss-Manin derivative in the tangent direction $w$: \begin{align*} \nabla_w\Omega_{\bullet}\big|_0=a_w\Omega+\eta_w. \end{align*} The polarized variation of Hodge structure on $R^n\pi_*\mathbb C$ has Hodge line $F^n$ and quotient horizontal component $E^{n-1}=F^{n-1}/F^n$. Therefore [Griffiths Curvature Formula for Graded Hodge Bundles][citetheorem:9144], applied to the Hermitian Hodge line $F^n$, identifies the Chern curvature of $F^n$ in the directions $(v,\bar w)$ with the negative of the Hodge-metric pairing of the second fundamental form components. Since the theorem defines $g^{\mathrm{WP}}$ as the positive curvature Hermitian form, this sign convention gives \begin{align*} -\partial_v\partial_{\bar w}\log h\big|_0=\frac{(\eta_v,\eta_w)_{\mathrm{HR}}}{\|\Omega\|^2}. \end{align*} The denominator is $\|\Omega\|^2=h(0)$, and the numerator is the Hodge-Riemann Hermitian pairing of the second fundamental form components $\eta_v$ and $\eta_w$ using the same primitive convention fixed in the statement.[/step]

[guided]We need to justify the curvature identity rather than treat it as a formal slogan. The family $\pi:\mathcal X\to U$ is smooth, proper, and Kähler, and it is polarized by the fixed Kähler class. Hence the middle cohomology local system $R^n\pi_*\mathbb C$ carries a polarized variation of Hodge structure with flat bundle $\mathcal H^n$, Hodge filtration $F^\bullet$, and Gauss-Manin connection $\nabla$. By [Griffiths Transversality][citetheorem:9129], differentiating a section of $F^n$ in a holomorphic tangent direction lands in $F^{n-1}$. Applied to the chosen local frame $\Omega_{\bullet}$ and the tangent vector $v\in T_0S$, this gives \begin{align*} \nabla_v\Omega_{\bullet}\big|_0\in F^{n-1}H^n(X,\mathbb C). \end{align*} The Hodge decomposition at the central fibre splits $F^{n-1}H^n(X,\mathbb C)$ as $H^{n,0}(X)\oplus H^{n-1,1}(X)$. Since $H^{n,0}(X)$ is spanned by $\Omega$, there are unique $a_v\in\mathbb C$ and $\eta_v\in H^{n-1,1}(X)$ such that \begin{align*} \nabla_v\Omega_{\bullet}\big|_0=a_v\Omega+\eta_v. \end{align*} The same decomposition for the tangent vector $w\in T_0S$ defines unique $a_w\in\mathbb C$ and $\eta_w\in H^{n-1,1}(X)$ by \begin{align*} \nabla_w\Omega_{\bullet}\big|_0=a_w\Omega+\eta_w. \end{align*} Now apply [Griffiths Curvature Formula for Graded Hodge Bundles][citetheorem:9144] to the polarized variation of Hodge structure just described, with $p=n$. Its hypotheses are exactly the polarized variation data: a flat Gauss-Manin bundle, a Hodge filtration, a polarization, and the Hodge metric induced by that polarization. For the Hodge line $F^n$, the Higgs field is the second fundamental form obtained by projecting $\nabla\Omega_{\bullet}$ from $F^{n-1}$ to $F^{n-1}/F^n\cong H^{n-1,1}(X)$. In the directions $v$ and $w$, these projected components are precisely $\eta_v$ and $\eta_w$. The cited Griffiths formula gives the Chern curvature of the Hodge line as the negative of the Hodge-metric pairing of this second fundamental form. In the local frame $\Omega_{\bullet}$, the curvature Hermitian form of the Hodge metric is represented by $-\partial\bar\partial\log h$ under the convention fixed in the statement. Thus evaluating this curvature form on $(v,\bar w)$ gives $-\partial_v\partial_{\bar w}\log h|_0$, and the Griffiths sign identifies that quantity with the positive Hodge-Riemann pairing of the horizontal components. With the primitive Hodge-Riemann convention fixed in the statement, the curvature formula gives \begin{align*} -\partial_v\partial_{\bar w}\log h\big|_0=\frac{(\eta_v,\eta_w)_{\mathrm{HR}}}{h(0)}. \end{align*} Since $h(0)=\|\Omega\|^2$ by definition of the Hodge norm at the central fibre, this becomes \begin{align*} -\partial_v\partial_{\bar w}\log h\big|_0=\frac{(\eta_v,\eta_w)_{\mathrm{HR}}}{\|\Omega\|^2}. \end{align*} The logarithm is what removes the line component $a_v\Omega$: curvature measures the horizontal second fundamental form, not the change caused by rescaling the chosen frame of the line.[/guided]

[step:Identify the horizontal component with contraction by the Kodaira-Spencer tensor]The theorem statement supplies a smooth proper holomorphic family with central fibre $X$, its Gauss-Manin connection, and the Kodaira-Spencer identification of $T_0S$ with the deformation classes in $H^1(X,T_X)$. Hence [[Formula for the Infinitesimal Period Map](/theorems/9131)][citetheorem:9131] applies at $0$ to the weight-$n$ Hodge filtration. Under [[Kodaira-Spencer Correspondence](/theorems/9117)][citetheorem:9117], the vector $v$ corresponds to the class $[\varphi]\in H^1(X,T_X)$. Therefore the $H^{n-1,1}$ component of $\nabla_v\Omega_{\bullet}|_0$ is the Dolbeault cohomology class represented by \begin{align*} \varphi\lrcorner\Omega\in A^{n-1,1}(X). \end{align*} Thus \begin{align*} \eta_v=[\varphi\lrcorner\Omega]\in H^{n-1,1}(X). \end{align*} Under the same Kodaira-Spencer correspondence, the vector $w$ corresponds to $[\psi]\in H^1(X,T_X)$. Applying the infinitesimal period map formula to $w$ gives that the $H^{n-1,1}$ component of $\nabla_w\Omega_{\bullet}|_0$ is represented by \begin{align*} \psi\lrcorner\Omega\in A^{n-1,1}(X), \end{align*} and hence \begin{align*} \eta_w=[\psi\lrcorner\Omega]\in H^{n-1,1}(X). \end{align*} The contraction map is an isomorphism on cohomology by [[Hodge-Theoretic Tangent Space for Calabi-Yau Deformations](/theorems/9138)][citetheorem:9138], so this identification is compatible with the chosen [harmonic representatives](/theorems/2747).[/step]

[guided]We now identify the abstract horizontal part $\eta_v$ with a concrete differential form. The tangent vector $v\in T_0S$ is identified with a Kodaira-Spencer class in $H^1(X,T_X)$. By hypothesis, $\varphi\in A^{0,1}(X,T_X)$ is the $\omega$-harmonic representative of this class, so $[\varphi]\in H^1(X,T_X)$ is the Kodaira-Spencer class of $v$. The derivative of the period map is described by contraction. The hypotheses needed for [Formula for the Infinitesimal Period Map][citetheorem:9131] are present here: the deformation is represented by a smooth proper holomorphic family, the bundle $\mathcal H^n$ has its Gauss-Manin connection, and $F^\bullet$ is the Hodge filtration of the fibres. The theorem says that the infinitesimal change of the Hodge filtration in the direction $v$ is obtained by applying the Kodaira-Spencer class to the original Hodge class. In the present top Hodge piece, this means that the component of $\nabla_v\Omega_{\bullet}|_0$ in $H^{n-1,1}(X)$ is represented by the $(n-1,1)$-form \begin{align*} \varphi\lrcorner\Omega\in A^{n-1,1}(X). \end{align*} Thus the class $\eta_v$ appearing in the Hodge decomposition is \begin{align*} \eta_v=[\varphi\lrcorner\Omega]\in H^{n-1,1}(X). \end{align*} For the vector $w\in T_0S$, the Kodaira-Spencer class is $[\psi]\in H^1(X,T_X)$ by the choice of the $\omega$-harmonic representative $\psi\in A^{0,1}(X,T_X)$. Applying the infinitesimal period map formula in the tangent direction $w$ gives that the $H^{n-1,1}$ component of $\nabla_w\Omega_{\bullet}|_0$ is represented by $\psi\lrcorner\Omega$. Therefore \begin{align*} \eta_w=[\psi\lrcorner\Omega]\in H^{n-1,1}(X). \end{align*} Finally, [Hodge-Theoretic Tangent Space for Calabi-Yau Deformations][citetheorem:9138] states that contraction with the holomorphic volume form identifies $H^1(X,T_X)$ with $H^{n-1,1}(X)$. This verifies that the contraction forms represent exactly the horizontal first variations used in the curvature computation, not merely some auxiliary representatives.[/guided]

[step:Convert the primitive Hodge-Riemann pairing into the displayed integral]Let $L_{\kappa}:H^n(X,\mathbb C)\to H^{n+2}(X,\mathbb C)$ denote cup product with the polarization class $\kappa$. The locally constant polarization class remains of type $(1,1)$ along the polarized deformation. Since $\Omega_s\in H^{n,0}(X_s)$, the class $[\Omega_s]$ is primitive for every nearby $s$, because wedging a class of type $(n,0)$ with a $(1,1)$ class has type $(n+1,1)$ and this Hodge component is zero on an $n$-dimensional complex manifold. Thus $L_{\kappa}[\Omega_s]=0$ along the family after flat identification of cohomology. Differentiating this identity in the tangent directions $v$ and $w$ gives $L_{\kappa}\eta_v=0$ and $L_{\kappa}\eta_w=0$, because the line components $a_v\Omega$ and $a_w\Omega$ are already primitive. Hence $\eta_v$ and $\eta_w$ lie in the primitive subspace of $H^{n-1,1}(X)$. Let $\mathcal I_X:=\mathcal I_0:A^{n,n}(X)\to\mathbb C$ denote the top-form integration functional on the central fibre. Since $\eta_v=[\varphi\lrcorner\Omega]$ and $\eta_w=[\psi\lrcorner\Omega]$, the primitive Hodge-Riemann convention fixed in the statement gives \begin{align*} (\eta_v,\eta_w)_{\mathrm{HR}}=(-1)^{n(n-1)/2}i^{n-2}\mathcal I_X((\varphi\lrcorner\Omega)\wedge\overline{(\psi\lrcorner\Omega)}). \end{align*} Also, by definition of the Hodge norm of the period line, \begin{align*} \|\Omega\|^2=i^{n^2}\mathcal I_X(\Omega\wedge\overline{\Omega}). \end{align*} Substituting these two identities into the curvature identity gives \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*} This is precisely the claimed Weil-Petersson pairing formula, and the equivalent Hodge-Riemann formulation follows from the same substitution.[/step]

[guided]The last step is only a substitution, but it is important to check that the pairing convention is being applied to the correct classes. Let $L_{\kappa}:H^n(X,\mathbb C)\to H^{n+2}(X,\mathbb C)$ denote cup product with the polarization class $\kappa$. Because the deformation is polarized, $\kappa$ is locally constant under the Gauss-Manin identification. For each nearby fibre, the class $[\Omega_s]$ has type $(n,0)$, so $L_{\kappa}[\Omega_s]$ has type $(n+1,1)$ and is zero on a complex manifold of dimension $n$. Therefore $L_{\kappa}[\Omega_s]=0$ along the family. Differentiating the identity $L_{\kappa}[\Omega_s]=0$ in the tangent direction $v$ gives $L_{\kappa}(a_v\Omega+\eta_v)=0$. The line component $a_v\Omega$ is primitive for the same type reason, so $L_{\kappa}\eta_v=0$. The same argument with the tangent direction $w$ gives $L_{\kappa}\eta_w=0$. Hence the two horizontal classes lie in the primitive subspace of $H^{n-1,1}(X)$, so the primitive Hodge-Riemann convention from the statement applies. Since the period-map computation gave \begin{align*} \eta_v=[\varphi\lrcorner\Omega] \end{align*} and \begin{align*} \eta_w=[\psi\lrcorner\Omega], \end{align*} the stated Hodge-Riemann convention gives \begin{align*} (\eta_v,\eta_w)_{\mathrm{HR}}=(-1)^{n(n-1)/2}i^{n-2}\mathcal I_X((\varphi\lrcorner\Omega)\wedge\overline{(\psi\lrcorner\Omega)}). \end{align*} The Hodge norm of the top Hodge line at the central fibre is \begin{align*} \|\Omega\|^2=i^{n^2}\mathcal I_X(\Omega\wedge\overline{\Omega}). \end{align*} Substituting these two identities into \begin{align*} g^{\mathrm{WP}}_0(v,w)=\frac{(\eta_v,\eta_w)_{\mathrm{HR}}}{\|\Omega\|^2} \end{align*} gives \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*} This proves the displayed Weil-Petersson formula, and the normalized Hodge-Riemann formulation is exactly the same identity before expanding the numerator and denominator.[/guided]