[proofplan] We identify the differential of the period map with the second fundamental form of the Hodge filtration under the Gauss-Manin connection. After restricting the family to a one-parameter slice tangent to $v$, the Kodaira-Spencer class is represented by the first-order Beltrami differential describing the change of complex structure on the fixed smooth fibre. A local relative form representing a Hodge class is then differentiated in this varying type decomposition, and its first new graded component is precisely contraction by the Beltrami differential. Finally we check that the resulting class is independent of all representatives, so the formula descends to Dolbeault cohomology. [/proofplan]

[step:Reduce the computation to a one-parameter deformation tangent to $v$]Let $v\in T_sS$. Choose a holomorphic map \begin{align*} \gamma:(\Delta,0)\to (S,s) \end{align*} from a small disc $\Delta\subset\mathbb C$ such that $d\gamma_0(\partial_t)=v$, where $t$ is the standard holomorphic coordinate on $\Delta$ and $\partial_t$ denotes the tangent vector at $0$. Form the pulled-back family \begin{align*} \pi_\Delta:\mathcal X_\Delta:=\mathcal X\times_S\Delta\to\Delta. \end{align*} Since $\pi$ is a smooth proper holomorphic submersion with Kähler fibres, the pulled-back map $\pi_\Delta$ has the same properties, and its central fibre is canonically $X$. By functoriality of the Kodaira-Spencer map, the Kodaira-Spencer class of $\pi_\Delta$ at $0$ is \begin{align*} \rho_\Delta(\partial_t)=\rho_s(v)\in H^1(X,T_X). \end{align*} By functoriality of the period map under base change, the derivative of $\Phi\circ\gamma$ at $0$ is $d\Phi_s(v)$. Therefore it is enough to prove the stated formula when the base is a disc, the point is $0$, and the tangent vector is $\partial_t$.[/step]

[guided]The differential $d\Phi_s(v)$ is first-order information in the single direction $v$. To compute such first-order information, we may replace the whole base $S$ by a one-dimensional slice through $s$ tangent to $v$. Choose a holomorphic map \begin{align*} \gamma:(\Delta,0)\to (S,s) \end{align*} from a small complex disc such that $d\gamma_0(\partial_t)=v$. Pull back the family: \begin{align*} \pi_\Delta:\mathcal X_\Delta:=\mathcal X\times_S\Delta\to\Delta. \end{align*} The hypotheses needed for Hodge theory are preserved: smoothness and properness are stable under base change, holomorphicity is inherited from $\pi$, and the fibres remain compact Kähler because they are fibres of the original Kähler family. There are two functoriality facts being used. First, the Kodaira-Spencer class of the pulled-back family in the tangent direction $\partial_t$ is the pullback of the original Kodaira-Spencer class: \begin{align*} \rho_\Delta(\partial_t)=\rho_s(d\gamma_0(\partial_t))=\rho_s(v). \end{align*} Second, the local period map for the pulled-back family is $\Phi\circ\gamma$, so its derivative at $0$ is \begin{align*} d(\Phi\circ\gamma)_0(\partial_t)=d\Phi_s(v). \end{align*} Thus proving the formula for the disc family proves it for the original family in the chosen direction.[/guided]

[step:Represent the Kodaira-Spencer class by the first-order Beltrami differential] Work over the disc. By Ehresmann's differentiable triviality theorem for proper submersions, after shrinking $\Delta$ there is a smooth trivialisation \begin{align*} \tau:X\times\Delta\to\mathcal X_\Delta \end{align*} over $\Delta$ whose restriction over $0$ is the identity on $X$. Transport the complex structure of the fibre $X_t:=\pi_\Delta^{-1}(t)$ to the fixed smooth manifold $X$, and denote the resulting complex structure by $J_t$. Thus $J_0$ is the original complex structure on $X$. The first-order variation of the subbundle $T^{0,1}_{J_t}X\subset (TX\otimes_{\mathbb R}\mathbb C)$ can be written uniquely in the form \begin{align*} T^{0,1}_{J_t}X=\{ \xi+t\,\mu(\xi)+O(|t|^2):\xi\in T^{0,1}_{J_0}X\}, \end{align*} where \begin{align*} \mu\in A^{0,1}(X,T_X) \end{align*} is a smooth $(0,1)$-form with values in the holomorphic tangent bundle $T_X=T^{1,0}_{J_0}X$. Integrability of the complex structures $J_t$ implies \begin{align*} \bar\partial\mu=0. \end{align*} With the sign convention in the statement, the first-order Beltrami differential is the tensor $\mu$ characterized by the expansion above, equivalently by the rule that a local $J_t$-holomorphic covector has fixed $J_0$-type expansion $\zeta-t\,\mu\lrcorner\zeta+O(|t|^2)$. This convention is part of the normalization of the Kodaira-Spencer map in the theorem statement. By the definition of the Kodaira-Spencer class for a one-parameter deformation, and equivalently by the [Kodaira-Spencer correspondence](/theorems/9117) [citetheorem:9117], the Dolbeault cohomology class $[\mu]\in H^1(X,T_X)$ is \begin{align*} [\mu]=\rho_\Delta(\partial_t)=\rho_s(v). \end{align*} [/step]

[step:Identify the infinitesimal period map with the second fundamental form] Let \begin{align*} \mathcal H^k:=R^k(\pi_\Delta)_*\mathbb C\otimes_{\mathbb C}\mathcal O_\Delta \end{align*} be the holomorphic vector bundle associated with the flat cohomology local system, let \begin{align*} \nabla:\mathcal H^k\to \mathcal H^k\otimes\Omega^1_\Delta \end{align*} be the Gauss-Manin connection, and let $F^\bullet\mathcal H^k$ be the Hodge filtration. The local marking identifies the period map with the map sending $t$ to the decreasing filtration $F^\bullet H^k(X_t,\mathbb C)$. For each $p$, the local marking trivializes the flat bundle and realizes the period map as a holomorphic map from $\Delta$ to the flag variety of filtrations on the fixed [vector space](/page/Vector%20Space) $H^k(X,\mathbb C)$. Differentiating a moving subspace $F^p_t\subset H^k(X,\mathbb C)$ in that flag variety gives its second fundamental form: lift a class in $F^p_0/F^{p+1}_0$ to a local section of $F^p\mathcal H^k$, differentiate in the flat trivialization, and take the result modulo $F^p_0$. Thus the component of the derivative of the period map on the graded piece is \begin{align*} \theta_p(\partial_t):F^p_0/F^{p+1}_0\to F^{p-1}_0/F^p_0 \end{align*} defined as follows: if $\sigma$ is a local holomorphic section of $F^p\mathcal H^k$ with $\sigma(0)$ representing a class in $F^p_0/F^{p+1}_0$, then \begin{align*} \theta_p(\partial_t)([\sigma(0)])=[\nabla_{\partial_t}\sigma(0)]\mod F^p_0. \end{align*} [Griffiths transversality](/theorems/9129) [citetheorem:9129] gives \begin{align*} \nabla_{\partial_t}(F^p\mathcal H^k)\subset F^{p-1}\mathcal H^k, \end{align*} so this expression lands in $F^{p-1}_0/F^p_0$. Under the Kähler [Hodge decomposition](/theorems/2745) this quotient is identified with $H^{p-1,q+1}(X)$ when $p+q=k$. [/step]

[step:Compute the first-order type change of a section of the Hodge filtration]Let $[\alpha]\in H^{p,q}(X)$, and choose a Dolbeault-closed representative \begin{align*} \alpha\in A^{p,q}(X) \end{align*} with $\bar\partial\alpha=0$. Choose a local holomorphic section $\sigma$ of $F^p\mathcal H^k$ whose value at $0$ maps to $[\alpha]$ in $F^p_0/F^{p+1}_0$. Using the smooth trivialisation, represent $\sigma(t)$ by a smooth family of closed complex $k$-forms \begin{align*} \eta_t\in A^k(X;\mathbb C) \end{align*} such that $\eta_0$ has fixed $J_0$-type $(p,q)$ component $\alpha$ modulo forms of type $(r,k-r)$ with $r\ge p+1$. Since $\sigma(t)\in F^p_t$, the fixed $J_t$-type decomposition of $\eta_t$ has no component with holomorphic degree less than $p$. For a smooth $k$-form $\omega$ on $X$, write $\operatorname{pr}^{a,b}_{J_0}\omega$ for its projection to the fixed $J_0$-type summand $A^{a,b}(X)$. The convention in the statement says that a local $J_t$-holomorphic covector has fixed $J_0$-type expansion $\zeta-t\,\mu\lrcorner\zeta+O(|t|^2)$. Wedge-multiplying this expansion over the $p$ holomorphic covector slots of the $(p,q)$ part of $\eta_0$ shows that replacing exactly one holomorphic slot contributes the coefficient $-\mu\lrcorner\alpha$ in fixed type $(p-1,q+1)$. Components of $\eta_0$ with holomorphic degree at least $p+1$ cannot contribute to fixed type $(p-1,q+1)$ to first order, because changing one holomorphic covector lowers the holomorphic degree by only one. If the local lift $\sigma$ or the representative form $\eta_t$ is changed while keeping the same section of $F^p\mathcal H^k$, the induced change in the de Rham derivative represents zero in cohomology; its fixed $(p-1,q+1)$ component is therefore Dolbeault-exact after passing to $F^{p-1}_0/F^p_0$. Hence \begin{align*} \operatorname{pr}^{p-1,q+1}_{J_0}(\partial_t\eta_t|_{t=0})=-\mu\lrcorner\alpha \end{align*} in $H^{q+1}(X,\Omega_X^{p-1})$. By the definition of the second fundamental form, \begin{align*} \theta_p(\partial_t)([\alpha])=[\nabla_{\partial_t}\sigma(0)]\mod F^p_0. \end{align*} Under the de Rham realization of the Gauss-Manin connection, this derivative is represented by $\partial_t\eta_t|_{t=0}$, and its image in $F^{p-1}_0/F^p_0\cong H^{p-1,q+1}(X)$ is represented by the fixed-type component computed above. Hence \begin{align*} \theta_p(\partial_t)([\alpha])=-[\mu\lrcorner\alpha]\in H^{q+1}(X,\Omega_X^{p-1}). \end{align*}[/step]

[guided]The second fundamental form must be computed from a section of the moving Hodge filtration, not from an arbitrary flat family. We therefore choose a local holomorphic section $\sigma$ of $F^p\mathcal H^k$ whose value at $0$ represents the given class $[\alpha]\in F^p_0/F^{p+1}_0\cong H^{p,q}(X)$. Using the smooth trivialisation, represent $\sigma(t)$ by closed complex $k$-forms \begin{align*} \eta_t\in A^k(X;\mathbb C) \end{align*} on the fixed smooth manifold $X$. At $t=0$, the class of the fixed $J_0$-type $(p,q)$ component of $\eta_0$ is represented by the chosen Dolbeault-closed form $\alpha\in A^{p,q}(X)$. Because $\sigma(t)$ lies in $F^p_t$, the form $\eta_t$ has no $J_t$-type component with holomorphic degree less than $p$. The first possible component below $F^p_0$ after expanding in the fixed $J_0$ decomposition is therefore the component of type $(p-1,q+1)$. The Beltrami differential \begin{align*} \mu\in A^{0,1}(X,T_X) \end{align*} encodes precisely this first-order comparison of type decompositions. For a smooth $k$-form $\omega$ on $X$, let $\operatorname{pr}^{a,b}_{J_0}\omega$ denote the projection of its fixed $J_0$-type decomposition onto $A^{a,b}(X)$. With the convention in the statement, a local $J_t$-holomorphic covector has fixed $J_0$-type expansion $\zeta-t\,\mu\lrcorner\zeta+O(|t|^2)$. Wedge-multiplying over the holomorphic covector slots of the $(p,q)$ component, the terms contributing to fixed type $(p-1,q+1)$ are exactly those in which one holomorphic covector is replaced by its first-order antiholomorphic correction. Each such replacement carries the coefficient from the displayed expansion, namely the minus sign. Thus \begin{align*} \operatorname{pr}^{p-1,q+1}_{J_0}(\partial_t\eta_t|_{t=0})=-\mu\lrcorner\alpha \end{align*} in Dolbeault cohomology. The remaining possible terms do not affect this cohomology class. Higher Hodge components of $\eta_0$ have holomorphic degree at least $p+1$, so one first-order replacement can lower them only to holomorphic degree at least $p$, which vanishes in the quotient $F^{p-1}_0/F^p_0$. Changing the local lift or the closed representative changes the de Rham derivative by a cohomologically zero term; after projecting to fixed type $(p-1,q+1)$ this gives a Dolbeault-exact form. Therefore the only surviving class in the quotient is $-[\mu\lrcorner\alpha]$. Now apply the definition of the second fundamental form. It is \begin{align*} \theta_p(\partial_t)([\alpha])=[\nabla_{\partial_t}\sigma(0)]\mod F^p_0. \end{align*} In the de Rham realization of the Gauss-Manin connection, $\nabla_{\partial_t}\sigma(0)$ is represented by $\partial_t\eta_t|_{t=0}$. Passing to the quotient $F^{p-1}_0/F^p_0$ keeps exactly the fixed $J_0$-type $(p-1,q+1)$ component. Therefore \begin{align*} \theta_p(\partial_t)([\alpha])=-[\mu\lrcorner\alpha]\in H^{q+1}(X,\Omega_X^{p-1}). \end{align*} This proves the contraction formula at the level of the second fundamental form.[/guided]

[step:Show the contraction formula is independent of representatives] We first check that $\mu\lrcorner\alpha$ is Dolbeault-closed. Since $\bar\partial\mu=0$ and $\bar\partial\alpha=0$, the Leibniz rule for the Dolbeault differential acting on the contraction pairing gives \begin{align*} \bar\partial(\mu\lrcorner\alpha)=0. \end{align*} Thus $\mu\lrcorner\alpha$ defines a class in $H^{q+1}(X,\Omega_X^{p-1})$. If $\alpha$ is replaced by $\alpha+\bar\partial\beta$, where \begin{align*} \beta\in A^{p,q-1}(X), \end{align*} then \begin{align*} \mu\lrcorner(\alpha+\bar\partial\beta)-\mu\lrcorner\alpha=\mu\lrcorner\bar\partial\beta. \end{align*} Using $\bar\partial\mu=0$ and the same Leibniz rule, the right-hand side is Dolbeault-exact up to the sign dictated by the contraction convention: \begin{align*} \mu\lrcorner\bar\partial\beta=\pm\bar\partial(\mu\lrcorner\beta). \end{align*} Therefore the cohomology class $[\mu\lrcorner\alpha]$ depends only on $[\alpha]$. If $\mu$ is replaced by $\mu+\bar\partial\xi$, where \begin{align*} \xi\in A^{0,0}(X,T_X), \end{align*} then \begin{align*} (\mu+\bar\partial\xi)\lrcorner\alpha-\mu\lrcorner\alpha=(\bar\partial\xi)\lrcorner\alpha. \end{align*} Since $\bar\partial\alpha=0$, the Leibniz rule gives \begin{align*} (\bar\partial\xi)\lrcorner\alpha=\pm\bar\partial(\xi\lrcorner\alpha), \end{align*} so the resulting Dolbeault cohomology class is unchanged. Hence the class $[\mu\lrcorner\alpha]$ depends only on $[\mu]=\rho_s(v)$ and $[\alpha]$. [/step]

[step:Conclude the infinitesimal period map formula] From the computation of the second fundamental form, \begin{align*} d\Phi_s(v)_{p,q}([\alpha])=\theta_p(v)([\alpha])=-[\mu\lrcorner\alpha]. \end{align*} By the identification $[\mu]=\rho_s(v)$, this is precisely \begin{align*} d\Phi_s(v)_{p,q}([\alpha])=-[\rho_s(v)\lrcorner\alpha]. \end{align*} The target is $H^{p-1,q+1}(X)$, equivalently $H^{q+1}(X,\Omega_X^{p-1})$, and the preceding step shows that the formula is well-defined on cohomology. This proves the stated formula for the infinitesimal period map. [/step]