Hodge-Riemann relations hold for large positive deformation, and the deformation principle

The basis and cross blocks. Assume $W\mathrm{\ne }0$; the zero case has zero tensor product and is vacuous. For $i\ge 0$, choose lifts $e_a\in B^{-i-1}$ of a basis of $W^{-i-1}$ orthogonal for its Lefschetz form, and lifts $p_b\in B^{-i+1}$ of a basis of $P_W^{-i+1}$; for $i=0$ the latter space is zero. Primitive decomposition (and $A:W^{-1}\to W^1$ when $i=0$) gives a basis $A{e}_a,p_b$ of $W^{-i+1}$. These lifts can be included in a homogeneous right basis of $B$, so $\alpha (A{e}_a),\beta (e_a),\alpha (p_b)$ reduce to a basis of $V^{-i}$, where $V=\overline{B{B}_s}$. By [F1]–[F2], $L_{\zeta }\beta (y)=\beta (Ay)$ after reduction, the $\beta$–$\beta$ Lefschetz block is zero, and $⟨\alpha (x),L_{\zeta }^i\beta (y)⟩=⟨x,A^{i}y{⟩}_W$. Consequently the $\alpha (Ae)$–$\beta (e)$ block is the invertible diagonal matrix $J=(⟨e_a,A^{i+1}{e}_b{⟩}_W)$, while the $\alpha (p)$–$\beta (e)$ block is zero, since $A^{i}p=0$ for primitives in degree $-i+1$.

The signature comparison. For nonzero $W$, let its lowest degree be $-d$ and put $b_i=\dim W^{-i}$ for integer $i$, so $b_{-i}=b_i$. For any $V$ as in the comparison statement put $v_i=\dim V^{-i}=b_{i-1}+b_{i+1}$. Under hL, its primitive dimensions are $p_i^V=v_i-v_{i+2}=b_{i-1}-b_{i+3}$, and $p_i^W=b_i-b_{i+2}$ for $i\ge 0$. The standard-HR signature required on $V^{-i}$ is the finite sum ${\sum }_{k\ge 0}(-1{)}^{(d+1-i-2k)\mathrm{/}2}{p}_{i+2k}^V$, in the occurring parity. Substitution and cancellation give $(-1{)}^{(d+1-i)\mathrm{/}2}(b_{i-1}-b_{i+1})$ for $i\ge 1$, exactly the signature of the primitive form on $P_W^{-i+1}$. For $i=0$ it is zero, since $b_{-1}=b_1$, matching $P_W^1=0$. By [F3] these signatures, together with hL, are equivalent to standard HR on $V$, proving both directions of the comparison. If $W=0$, the dimension identity forces $V=0$ and both sides are vacuous.

The hyperbolic blocks. The Lefschetz matrix on $V^{-i}$ therefore has form $\left(\begin{array}{ccc}C& J& E\\ J& 0& 0\\ E^{\mathsf{T}}& 0& Q_{\zeta }\end{array}\right)$. Subtracting suitable vectors from the $\beta$ block from the $\alpha (p)$ block kills $E$ without changing $Q_{\zeta }$; replacing the first block by itself minus one half of the corresponding $J^{-1}C$ combination of the second kills $C$. This gives a congruence to $\left(\begin{array}{ccc}0& J& 0\\ J& 0& 0\\ 0& 0& Q_{\zeta }\end{array}\right)$. The first two blocks are non-degenerate with equal positive and negative inertia. Thus the full matrix is non-degenerate exactly when $Q_{\zeta }$ is, and its signature is that of $Q_{\zeta }$. For $i=0$, $Q_{\zeta }$ is empty, so the matrix is already non-degenerate of signature zero for every parameter.

The primitive block for $i>0$. For the chosen lifts $p,q$ of degree $-i+1$, binomial expansion and [F1] give $⟨\alpha (p),L_{\zeta }^i\alpha (q)⟩={\sum }_{j=0}^i\left(\genfrac{}{}{0px}{}{i}{j}\right){\zeta }^j{\mathrm{\partial }}_s(⟨p,A^{i-j}q{⟩}_B{\rho }^j)$ before reduction. The polynomial $⟨p,A^{i-j}q{⟩}_B$ has degree $2-2j$: it vanishes for $j\ge 2$, and for $j=1$ it is constant. Since ${\mathrm{\partial }}_s\rho =\rho ({\alpha }_s^{\vee })>0$, reduction yields $Q_{\zeta }=Q_0+\zeta i\rho ({\alpha }_s^{\vee })Q$, where $Q$ is the definite primitive Lefschetz matrix on $P_W^{-i+1}$. If this space is zero the block is empty. Otherwise $i>0$ and ${\zeta }^{-1}{Q}_{\zeta }$ tends to the definite matrix $i\rho ({\alpha }_s^{\vee })Q$; symmetric elimination near that matrix preserves the pivot signs by [F3]. Hence for sufficiently large $\zeta$, $Q_{\zeta }$ is definite with the signature of $Q$. There are only finitely many nonzero graded pieces, so one bound on $\zeta$ works for all of them.

Large parameters. Steps 2.1–3.1 make every Lefschetz form on $V^{-i}$ non-degenerate for all sufficiently large parameters, so [F3] gives hL. Their signatures are precisely the primitive signatures of $W$; step 1.2 gives HR with the standard sign on $V$. This proves the large-parameter assertion, including the empty primitive blocks and $i=0$.

Deformation principle. On any compact interval joining ${\zeta }_0$ to another nonnegative parameter, the polynomial family $L_{\zeta }$ is continuous. If hL holds throughout, [F3] transfers HR from ${\zeta }_0$ to that parameter. When HR$(x)$ holds, step 4.1 supplies a large positive ${\zeta }_0$; otherwise the more general corollary assumes one such parameter. This proves both corollaries and completes the argument without choice.