The middle primitive polarization follows from the nonmiddle forms and a Kähler limit

Put $d=b_n-b_{n-2}$. By [F1] each map $L+ϵ\eta :H^n\to H^{n+2}$ has kernel of dimension $d$, and these kernels are Hodge subspaces by [F1]. The matrix of the map is polynomial in $ϵ$; choose a maximal minor that is nonzero on a sufficiently small punctured interval $(0,{ϵ}_0)$. On that interval row elimination expresses a basis of the kernel ${\mathrm{\Lambda }}_{ϵ}$ by rational functions of $ϵ$; wedge the $d$ basis vectors, clear denominators, and divide by the lowest power of $ϵ$ occurring. The result is a nonzero polynomial exterior vector whose value at $ϵ=0$ defines a point $\mathrm{\Lambda }$ of the Grassmannian $\mathrm{G}(d,H^n)$, and $\mathrm{\Lambda }={\lim }_{ϵ\downarrow 0}{\mathrm{\Lambda }}_{ϵ}$: the Plücker coordinates of ${\mathrm{\Lambda }}_{ϵ}$ are given by that rational exterior vector, so they converge to the value at $0$. A different maximal minor gives the same punctured family of planes and hence the same limit point. Stability of a plane under a fixed endomorphism of $H^n$ is a closed condition on the Grassmannian, so $\mathrm{\Lambda }$ is stable under every Hodge projector, being a limit of the Hodge subspaces ${\mathrm{\Lambda }}_{ϵ}$; hence $\mathrm{\Lambda }$ is a real Hodge subspace. If $u_{ϵ}\in {\mathrm{\Lambda }}_{ϵ}$ and $u_{ϵ}\to u$, then $L{u}_{ϵ}=-ϵ\eta u_{ϵ}\to 0$ by continuity, so $Lu=0$ and $\mathrm{\Lambda }\subseteq D$.

Put $E_j:={\eta }^j\ker (L^j:H^{n-2j}\to H^n)$, so $M=E_1$. We prove by induction on $i$ that $M\cap E_1^{\perp }\cap \cdots \cap E_i^{\perp }=M\cap W_{-i}{H}^n\mathrm{.}$ For $i=0$ both sides equal $M$, because $\ker L\subseteq W_0$ by [F2] and $\eta$ preserves $W$ by (a), so $M\subseteq W_0$.

Induction hypothesis: fix $i\ge 0$ and assume the displayed equality $M\cap E_1^{\perp }\cap \cdots \cap E_i^{\perp }=M\cap W_{-i}{H}^n$; we must derive the equality for $i+1$. Since $\ker L^{i+1}\subseteq W_i$ by [F2] and $\eta$ preserves $W$, one has $E_{i+1}\subseteq W_i$; condition (c) (with $a=-i-1$, $b=i$ and $a+b=-1<0$) therefore makes $M\cap W_{-i-1}{H}^n$ orthogonal to $E_{i+1}$, and so $M\cap W_{-i-1}{H}^n\subseteq M\cap E_1^{\perp }\cap \cdots \cap E_i^{\perp }\cap E_{i+1}^{\perp }$ by the induction hypothesis. This is one inclusion of the equality for $i+1$.

For the converse inclusion let $\alpha =\eta \lambda \in M\cap W_{-i}{H}^n$ be orthogonal to $E_{i+1}$, where $L\lambda =0$. First we show $\lambda \in W_{-i}{H}^{n-2}$. Otherwise $\lambda \in W_0$ has a largest graded weight $a$ with $-i<a\le 0$ and $[\lambda {]}_a\mathrm{\ne }0$ in ${\mathrm{Gr}}_a^W{H}^{n-2}$; since $\alpha =\eta \lambda \in W_{-i}$ and $-i<a$, the graded class $[\alpha {]}_a=0$, while $[\alpha {]}_a=\eta [\lambda {]}_a$ because $\eta$ preserves $W$; as $a\le 0$, hypothesis (a) makes $\eta :{\mathrm{Gr}}_a^W{H}^{n-2}\to {\mathrm{Gr}}_a^W{H}^n$ injective, a contradiction. Next, centered hard Lefschetz decomposes $\lambda$ along the graded pieces: because $L^i:{\mathrm{Gr}}_i^W\to {\mathrm{Gr}}_{-i}^W$ is an isomorphism, the ${\mathrm{Gr}}_{-i}^W$-class of $\lambda$ lifts to a class in ${\mathrm{Gr}}_i^W$ with representative ${\lambda }_1\in W_i{H}^{n-2-2i}$, and ${\tau }_1:=\lambda -L^i{\lambda }_1\in W_{-i-1}{H}^{n-2}$. Repeating with ${\tau }_1$ and $L^{i+1}$, then with the successive remainders and $L^{i+2},L^{i+3},\dots$, and using that $W$ is a finite filtration, we obtain $\lambda =L^i{\lambda }^{\mathrm{\prime }}$ with ${\lambda }^{\mathrm{\prime }}\in W_i{H}^{n-2-2i}$; indeed at stage $t$ the correction $L^{t-1}{\lambda }_t$ lies in $W_{i-(t-1)}{H}^{n-2-2i}$, so all corrections lie in $W_i$. Then $L^{i+1}{\lambda }^{\mathrm{\prime }}=L\lambda =0$. Every class of $\ker (L^{i+1}:{\mathrm{Gr}}_i^W{H}^{n-2-2i}\to {\mathrm{Gr}}_{i-2i-2}^W)$ has an actual lift in $W_i{H}^{n-2-2i}$ annihilated by $L^{i+1}$: for any lift the error $L^{i+1}{\lambda }^{\mathrm{\prime }\mathrm{\prime }}$ lies in $W_{-i-3}{H}^n$, and the homogeneous-string description of [F2] shows $L^{i+1}:W_{i-1}{H}^{n-2-2i}\to W_{-i-3}{H}^n$ is surjective, since on each string it carries the vector of weight $b+2i+2$ onto the vector of weight $b$ for every weight $b\le -i-3$; subtracting a preimage of the error from ${\lambda }^{\mathrm{\prime }\mathrm{\prime }}$ gives the required lift. Pairing the orthogonality of $\alpha$ to $E_{i+1}$ with these lifts ${\beta }^{\mathrm{\prime }}$ shows $0={\int }_X\alpha \wedge {\eta }^{i+1}{\beta }^{\mathrm{\prime }}={\int }_X{\eta }^{i+2}{L}^i{\lambda }^{\mathrm{\prime }}\wedge {\beta }^{\mathrm{\prime }}=q_i([{\lambda }^{\mathrm{\prime }}],[{\beta }^{\mathrm{\prime }}])$ for every class $[{\beta }^{\mathrm{\prime }}]$ in $\ker (L^{i+1}:{\mathrm{Gr}}_i^W{H}^{n-2-2i}\to {\mathrm{Gr}}_{i-2i-2}^W)$. Hypothesis (b) now forces $[{\lambda }^{\mathrm{\prime }}]=0$, that is, ${\lambda }^{\mathrm{\prime }}\in W_{i-1}{H}^{n-2-2i}$, and since $\eta$ preserves $W$ while $L$ shifts it by $-2$ one gets $\alpha =\eta L^i{\lambda }^{\mathrm{\prime }}\in W_{-i-1}{H}^n$. This proves the converse inclusion and completes the induction, so for $i$ beyond the finite filtration, $M\cap E_1^{\perp }\cap E_2^{\perp }\cap \cdots =0$.

Let $u_{ϵ}\in {\mathrm{\Lambda }}_{ϵ}$. Commutation of the two even-degree classes $\eta ,L$ gives ${\eta }^j{u}_{ϵ}=(-{ϵ}^{-1}{)}^j{L}^j{u}_{ϵ}$, hence ${\int }_X{u}_{ϵ}\wedge {\eta }^{j}z=(-{ϵ}^{-1}{)}^j{\int }_X{u}_{ϵ}\wedge L^{j}z=0$ for every $z$ with $L^{j}z=0$. Passing to the limit, $\mathrm{\Lambda }\subseteq E_j^{\perp }$ for every $j\ge 1$. By step 3.1, $\mathrm{\Lambda }\cap M=0$. The rank of $L:H^n\to H^{n+2}$ equals the rank of $L:H^{n-2}\to H^n$, because the two maps are adjoint under the Poincaré pairings of [F1]; moreover $\eta :H^{n-2}\to H^n$ is injective by [F1]. Therefore $\dim D=b_n-\mathrm{rk}(L:H^n\to H^{n+2})=(b_n-b_{n-2})+(b_{n-2}-\mathrm{rk}(L:H^{n-2}\to H^n\left)\right)=\dim \mathrm{\Lambda }+\dim M$, using $\dim \mathrm{\Lambda }=d=b_n-b_{n-2}$ from step 1.1. Since $\mathrm{\Lambda }\oplus M\subseteq D$ (both summands lie in $D$: $\mathrm{\Lambda }$ by step 1.1 and $M$ because $L\eta =\eta L$ kills $\eta \ker L$) and the dimensions agree, $D=\mathrm{\Lambda }\oplus M$; and because $\mathrm{\Lambda }\subseteq E_1^{\perp }$ while $M=E_1$, this sum is orthogonal. Moreover $\mathrm{\Lambda }\subseteq D\cap {\bigcap }_j{E}_j^{\perp }$ has dimension $\dim D-\dim M=\dim \mathrm{\Lambda }$ and $M\cap {\bigcap }_j{E}_j^{\perp }=0$, so the inclusion is an equality, giving the asserted characterization of $\mathrm{\Lambda }$.

Poincaré duality identifies the orthogonal complement of $D$ inside $H^n$ with the image of $L:H^{n-2}\to H^n$, by the adjointness used in step 4.1. On a homogeneous string the kernel of $L$ is the last vector $L^{\mathrm{\ell }-1}v$, of weight $1-\mathrm{\ell }$; this vector lies in the image of $L$ precisely when $\mathrm{\ell }>1$, equivalently when its weight is negative, so $D\cap \mathrm{im}L=D\cap W_{-1}{H}^n=R$ and by definition $\mathrm{rad}(D)=D\cap D^{\perp }=D\cap \mathrm{im}L=R$. An orthogonal decomposition of a finite-dimensional space with radical $R$ descends to an orthogonal decomposition of the nondegenerate quotient: if $x=x_{\mathrm{\Lambda }}+x_M\in R$ is paired with each summand separately, orthogonality of the decomposition and $x\in \mathrm{rad}$ show both components are radical, so $R=(R\cap \mathrm{\Lambda })\oplus (R\cap M)$ and the induced form on each quotient summand $\mathrm{\Lambda }\mathrm{/}(\mathrm{\Lambda }\cap R)$ and $M\mathrm{/}(M\cap R)$ is nondegenerate.

The classical Hermitian form $Q_n(Ca,\bar{a})$ of [F1] is positive on every ${\mathrm{\Lambda }}_{ϵ}$, because ${\mathrm{\Lambda }}_{ϵ}$ is the degree-$n$ primitive space of the Kähler class $L+ϵ\eta$. For $a\in \mathrm{\Lambda }$ choose $a_{ϵ}\in {\mathrm{\Lambda }}_{ϵ}$ with $a_{ϵ}\to a$; by continuity $Q_n(Ca,\bar{a})={\lim }_{ϵ\downarrow 0}{Q}_n(C{a}_{ϵ},{\bar{a}}_{ϵ})\ge 0$, so the form is semipositive on $\mathrm{\Lambda }$. The Hodge stability of $\mathrm{\Lambda }$ and of $R=D\cap W_{-1}$, together with $\eta$- and $L$-stability of $W$, makes $\mathrm{\Lambda }\mathrm{/}(\mathrm{\Lambda }\cap R)$ a Hodge quotient. Step 5.1 gives nondegeneracy of the induced form there, and a nondegenerate semipositive Hermitian form is positive definite. The Hodge–Riemann bilinear relations pass from the ambient degree-$n$ cohomology of [F1], so this restriction is a polarization with exactly the sign $ϵ(n)$ of [F1]. In $D\mathrm{/}R$ the orthogonal complement of the image of $M$ is the image of $\mathrm{\Lambda }$, since the decomposition of step 5.1 is orthogonal. Every Hodge subspace of that complement inherits the positive definite form, hence is polarized; this proves the final assertion, and in particular the middle double primitive subspace is polarized as soon as its graded primitive condition identifies it with such a Hodge subspace. The degenerate cases (empty or zero graded pieces, $d=0$, and the trivial string of length one with weight $0$) are covered by the same formulas, which then reduce to identities between zero-dimensional spaces.