[proofplan] The proof translates the infinitesimal variation of the primitive Hodge structure into multiplication in the Jacobian ring. Griffiths residue theory identifies the primitive Hodge piece $H^{n-q,q}(X)_{\mathrm{prim}}$ with the graded piece $R_{f,(q+1)d-n-2}$, and the infinitesimal period map in the direction represented by $a\in V\subset R_{f,d}$ becomes multiplication by $a$. Therefore a tangent vector killed by the primitive period differential has zero multiplication action on the specified graded piece. The assumed injectivity of the multiplication representation then forces the tangent vector to be zero. [/proofplan]

[step:Represent hypersurface deformation classes in the degree-$d$ Jacobian ring]Let \begin{align*} A:=\mathbb C[x_0,\dots,x_{n+1}] \end{align*} be the homogeneous coordinate ring of $\mathbb P^{n+1}$, and let \begin{align*} J_f:=\left(\frac{\partial f}{\partial x_0},\dots,\frac{\partial f}{\partial x_{n+1}}\right)\subset A \end{align*} be the Jacobian ideal of $f$. Thus $R_f=A/J_f$, and $R_{f,m}$ denotes the image of $A_m$ in $R_f$. By hypothesis, the Kodaira-Spencer map for the chosen local deformation space $B$ of smooth degree-$d$ hypersurfaces identifies its Zariski tangent space $T=T_XB$ with a linear subspace $V\subset R_{f,d}$. Therefore each tangent vector $\xi\in T$ has a uniquely determined class \begin{align*} a_\xi\in V\subset R_{f,d} \end{align*} representing the corresponding first-order degree-$d$ perturbation of the defining equation modulo infinitesimal projective automorphisms.[/step]

[guided]The deformation-theoretic input is that first-order motions of the hypersurface are represented by degree-$d$ perturbations of the defining equation, with perturbations coming from infinitesimal projective coordinate changes quotiented out. Algebraically, this quotient is encoded in the degree-$d$ part of the Jacobian ring. We have defined the homogeneous coordinate ring \begin{align*} A:=\mathbb C[x_0,\dots,x_{n+1}] \end{align*} and the Jacobian ideal \begin{align*} J_f:=\left(\frac{\partial f}{\partial x_0},\dots,\frac{\partial f}{\partial x_{n+1}}\right)\subset A. \end{align*} Thus $R_f=A/J_f$, and $R_{f,d}$ is the degree-$d$ quotient of $A_d$ by the degree-$d$ part of $J_f$. By the stated Kodaira-Spencer identification, the tangent space $T=T_XB$ is identified with a linear subspace $V\subset R_{f,d}$. Hence, for every tangent vector $\xi\in T$, there is a unique element \begin{align*} a_\xi\in V\subset R_{f,d} \end{align*} corresponding to $\xi$. This element is the Jacobian-ring class of the first-order degree-$d$ perturbation of $f$ represented by $\xi$, after quotienting by infinitesimal projective automorphisms.[/guided]

[step:Use Griffiths residues to identify primitive Hodge pieces with graded Jacobian-ring pieces]We use the Griffiths residue description of primitive cohomology for smooth complex projective hypersurfaces: since $X\subset\mathbb P^{n+1}$ is smooth of degree $d$, for each integer $p$ with $0\le p\le n$, residue induces an isomorphism of complex vector spaces \begin{align*} \operatorname{Res}_p:R_{f,(p+1)d-n-2}\to H^{n-p,p}(X)_{\mathrm{prim}}. \end{align*} Applying this with $p=q$ and $p=q+1$, and using $0\le q\le n-1$, gives isomorphisms \begin{align*} \operatorname{Res}_q:R_{f,(q+1)d-n-2}\to H^{n-q,q}(X)_{\mathrm{prim}} \end{align*} and \begin{align*} \operatorname{Res}_{q+1}:R_{f,(q+2)d-n-2}\to H^{n-q-1,q+1}(X)_{\mathrm{prim}}. \end{align*} Thus the source and target of the $q$-th Hodge component of the infinitesimal primitive period map may be expressed in the two displayed graded pieces of $R_f$.[/step]

[guided]The point of introducing the residue isomorphisms is to replace a Hodge-theoretic question by an algebraic one. The primitive period map records how the Hodge filtration on the primitive middle cohomology changes under deformation. Its $q$-th infinitesimal component lowers Hodge type from $(n-q,q)$ to $(n-q-1,q+1)$. The Griffiths residue theorem for smooth complex projective hypersurfaces applies because $X\subset\mathbb P^{n+1}$ is smooth and is cut out by a homogeneous polynomial of degree $d$. It says that, for every integer $p$ with $0\le p\le n$, there is an isomorphism \begin{align*} \operatorname{Res}_p:R_{f,(p+1)d-n-2}\to H^{n-p,p}(X)_{\mathrm{prim}}. \end{align*} Here $R_{f,(p+1)d-n-2}$ is the degree-$(p+1)d-n-2$ part of the Jacobian ring, and $H^{n-p,p}(X)_{\mathrm{prim}}$ is the primitive part of the Hodge summand in middle cohomology. We apply this theorem first with $p=q$. Since $0\le q\le n-1$, the integer $q$ lies in the allowed range, so we obtain \begin{align*} \operatorname{Res}_q:R_{f,(q+1)d-n-2}\to H^{n-q,q}(X)_{\mathrm{prim}}. \end{align*} We apply it again with $p=q+1$. The condition $q\le n-1$ ensures $q+1\le n$, so this second application is also in range and gives \begin{align*} \operatorname{Res}_{q+1}:R_{f,(q+2)d-n-2}\to H^{n-q-1,q+1}(X)_{\mathrm{prim}}. \end{align*} These two identifications are exactly the source and target Hodge pieces connected by the $q$-th component of the infinitesimal period map.[/guided]

[step:Identify the infinitesimal period map with multiplication by the deformation class]Let $D$ denote the classifying period domain of polarized Hodge filtrations on $H^n(X,\mathbb C)_{\mathrm{prim}}$ with the same Hodge numbers and polarization type as $X$, and let $d\mathcal P_X:T\to T_{\mathcal P(X)}D$ denote the differential at $X$ of the primitive period map. The deformation space $B$ parametrizes smooth projective hypersurfaces, so after shrinking $B$ it gives a smooth proper Kähler family whose primitive middle cohomology carries a polarized variation of Hodge structure. By [citetheorem:9129] and [citetheorem:9130], [Griffiths transversality](/theorems/9129) puts the differential of the local period map in the horizontal tangent space. The horizontal tangent space is the Griffiths-transverse part of the tangent space to $D$, so we view the horizontal component of $d\mathcal P_X$ as a map \begin{align*} d\mathcal P_X:T\to \bigoplus_{p=0}^{n-1}\operatorname{Hom}_{\mathbb C}\left(H^{n-p,p}(X)_{\mathrm{prim}},H^{n-p-1,p+1}(X)_{\mathrm{prim}}\right). \end{align*} For $\xi\in T$, let \begin{align*} (d\mathcal P_X)_q(\xi):H^{n-q,q}(X)_{\mathrm{prim}}\to H^{n-q-1,q+1}(X)_{\mathrm{prim}} \end{align*} be its $q$-th Hodge component. By [citetheorem:9131], the infinitesimal period map is cup product with the Kodaira-Spencer class followed by contraction of forms. We use this formula together with Griffiths' infinitesimal residue computation for smooth hypersurfaces: for first-order degree-$d$ deformations modulo the Jacobian ideal, this cup-product-and-contraction action on primitive cohomology is compatible with the residue isomorphisms and is given on the Jacobian ring by multiplication by the corresponding class. Therefore, under the residue identifications above and the Kodaira-Spencer identification of $\xi$ with $a_\xi\in R_{f,d}$, $(d\mathcal P_X)_q(\xi)$ is represented by multiplication by $a_\xi$. Equivalently, the following square commutes: \begin{align*} \operatorname{Res}_{q+1}^{-1}\circ (d\mathcal P_X)_q(\xi)\circ \operatorname{Res}_q=\mu_q(a_\xi). \end{align*} Thus, for every \begin{align*} b\in R_{f,(q+1)d-n-2}, \end{align*} the class corresponding to $(d\mathcal P_X)_q(\xi)(\operatorname{Res}_q(b))$ is \begin{align*} a_\xi b\in R_{f,(q+2)d-n-2}. \end{align*}[/step]

[guided]We now identify the Hodge-theoretic differential with the algebraic multiplication map. The infinitesimal period map has a component \begin{align*} (d\mathcal P_X)_q(\xi):H^{n-q,q}(X)_{\mathrm{prim}}\to H^{n-q-1,q+1}(X)_{\mathrm{prim}} \end{align*} for each tangent vector $\xi\in T$. By the Kodaira-Spencer identification in the statement, $\xi$ corresponds to a unique element \begin{align*} a_\xi\in V\subset R_{f,d}. \end{align*} The [formula for the infinitesimal period map](/theorems/9131) applies because the deformation over $B$ is a smooth proper Kähler family and $\xi\in T_XB$ is represented by its Kodaira-Spencer class. By [citetheorem:9131], the component is obtained by cup product with that Kodaira-Spencer class and contraction of forms. The theorem applies to the primitive summand because the primitive cohomology is a flat polarized sub-variation for a smooth projective hypersurface family. For smooth hypersurfaces, Griffiths' infinitesimal residue computation says more precisely that first-order degree-$d$ deformations modulo the Jacobian ideal act on the residue model by multiplication by their Jacobian-ring class, compatibly with the primitive residue identifications. With the residue isomorphisms $\operatorname{Res}_q$ and $\operatorname{Res}_{q+1}$ fixed in the previous step, the precise algebraic statement is \begin{align*} \operatorname{Res}_{q+1}^{-1}\circ (d\mathcal P_X)_q(\xi)\circ \operatorname{Res}_q=\mu_q(a_\xi). \end{align*} This equality is an identity of maps from $R_{f,(q+1)d-n-2}$ to $R_{f,(q+2)d-n-2}$. Consequently, if \begin{align*} b\in R_{f,(q+1)d-n-2}, \end{align*} then $\operatorname{Res}_q(b)$ is the corresponding primitive Hodge class of type $(n-q,q)$, and applying the $q$-th infinitesimal period component gives a class whose inverse residue image is \begin{align*} \mu_q(a_\xi)(b)=a_\xi b\in R_{f,(q+2)d-n-2}. \end{align*} This is the point where the geometric deformation problem becomes the injectivity problem for multiplication by $a_\xi$ in the Jacobian ring.[/guided]

[step:Use injectivity of multiplication to prove injectivity of the period differential]Let $\xi\in T$ satisfy \begin{align*} d\mathcal P_X(\xi)=0. \end{align*} Then its $q$-th component also vanishes: \begin{align*} (d\mathcal P_X)_q(\xi)=0. \end{align*} Using the multiplication description from the previous step, we obtain \begin{align*} \mu_q(a_\xi)=\operatorname{Res}_{q+1}^{-1}\circ (d\mathcal P_X)_q(\xi)\circ \operatorname{Res}_q=0. \end{align*} By the hypothesis that \begin{align*} \mu_q:V\to \operatorname{Hom}_{\mathbb C}\left(R_{f,(q+1)d-n-2},R_{f,(q+2)d-n-2}\right) \end{align*} is injective, $\mu_q(a_\xi)=0$ implies \begin{align*} a_\xi=0 \end{align*} in $V$. Since the Kodaira-Spencer identification identifies $\xi$ with $a_\xi$, it follows that \begin{align*} \xi=0. \end{align*} Therefore $\ker d\mathcal P_X=\{0\}$, so the differential of the primitive period map is injective at $X$.[/step]

[guided]Assume that a tangent vector $\xi\in T$ is killed by the differential of the primitive period map: \begin{align*} d\mathcal P_X(\xi)=0. \end{align*} Then every Hodge component of this horizontal tangent vector is zero, and in particular \begin{align*} (d\mathcal P_X)_q(\xi)=0. \end{align*} The previous step identified the $q$-th component with multiplication by the Jacobian-ring class $a_\xi$. Therefore \begin{align*} \mu_q(a_\xi)=\operatorname{Res}_{q+1}^{-1}\circ (d\mathcal P_X)_q(\xi)\circ \operatorname{Res}_q=0. \end{align*} Here the zero on the right is the zero map from $R_{f,(q+1)d-n-2}$ to $R_{f,(q+2)d-n-2}$. Now we use exactly the algebraic hypothesis in the theorem statement: the map \begin{align*} \mu_q:V\to \operatorname{Hom}_{\mathbb C}\left(R_{f,(q+1)d-n-2},R_{f,(q+2)d-n-2}\right) \end{align*} is injective. Since $a_\xi\in V$ and $\mu_q(a_\xi)=0$, injectivity forces \begin{align*} a_\xi=0. \end{align*} Finally, the Kodaira-Spencer identification between $T$ and $V$ sends $\xi$ to $a_\xi$. Thus $a_\xi=0$ implies \begin{align*} \xi=0. \end{align*} Every tangent vector in the kernel is zero, so $\ker d\mathcal P_X=\{0\}$ and the primitive period differential is injective at $X$.[/guided]

[step:Verify the stated standard examples] For a smooth plane curve, $n=1$. Taking $q=0$, the multiplication representation in the criterion is \begin{align*} \mu_0:V\to \operatorname{Hom}_{\mathbb C}\left(R_{f,d-3},R_{f,2d-3}\right), \end{align*} with $a\in V\subset R_{f,d}$ acting by $b\mapsto ab$. For smooth plane curves of degree $d\ge 4$, the standard Griffiths-Green Jacobian-ring multiplication theorem states precisely that multiplication by degree-$d$ deformation classes gives an injective map from the relevant Kodaira-Spencer subspace $V\subset R_{f,d}$ into $\operatorname{Hom}_{\mathbb C}(R_{f,d-3},R_{f,2d-3})$. For a smooth cubic threefold, $n=3$ and $d=3$. Taking $q=1$, the multiplication representation in the criterion is \begin{align*} \mu_1:V\to \operatorname{Hom}_{\mathbb C}\left(R_{f,1},R_{f,4}\right), \end{align*} again with $a\in V\subset R_{f,3}$ acting by $b\mapsto ab$. The corresponding cubic-threefold Jacobian-ring multiplication theorem states precisely that this map from the relevant Kodaira-Spencer subspace $V\subset R_{f,3}$ into $\operatorname{Hom}_{\mathbb C}(R_{f,1},R_{f,4})$ is injective. In both cases the Griffiths residue computation supplies the Hodge-theoretic identifications required in the statement, and the displayed multiplication injectivity is exactly the hypothesis of the criterion proved above. Hence the differential of the primitive period map is injective at $X$ for smooth plane curves of degree $d\ge 4$ and for smooth cubic threefolds. [/step]