Hodge–Riemann bilinear relations for smooth projective varieties

Let $a,b$ be closed smooth representatives of classes in $P^k$. Replacing $a$ by $a+du$ changes the integrand by $du\wedge b\wedge {\omega }^{n-k}$, which is $d(u\wedge b\wedge {\omega }^{n-k})$ up to sign because $b$ and $\omega$ are closed; the general Stokes theorem [F6] on the compact boundaryless $X$ gives ${\int }_{X}d(\cdot )=0$, and the same argument applies in the second variable, so $Q_k$ depends only on the classes. Since $\mathrm{deg}a=\mathrm{deg}b=k$, one has $b\wedge a=(-1{)}^{k}a\wedge b$, hence $Q_k(b,a)=(-1{)}^k{Q}_k(a,b)$. If $a$ has type $(p,q)$ and $b$ type $(r,s)$ with $p+q=r+s=k$, the top form $a\wedge b\wedge {\omega }^{n-k}$ has type $(p+r+n-k,q+s+n-k)$, which can equal $(n,n)$ only if $p+r=k=q+s$, that is $(r,s)=(q,p)$; so $Q_k$ pairs $P^{p,q}$ only with $P^{q,p}$. For nonzero $a\in P^{p,q}$ let $a$ also denote its harmonic representative, which is primitive of type $(p,q)$ by [F2, F3, F4] and satisfies the displayed positivity: the conjugate form $\bar{a}$ is primitive of type $(q,p)$, so the pointwise Weil star identity [F2] gives $\ast \bar{a}=ϵ(k)i^{p-q}{L}^{n-k}\bar{a}\mathrm{/}(n-k)!$, whence $a\wedge \ast \bar{a}=ϵ(k)i^{p-q}a\wedge \bar{a}\wedge {\omega }^{n-k}\mathrm{/}(n-k)!$ and, integrating with the $L^2$ pairing of [F1], $i^{p-q}{Q}_k(a,\bar{a})=(n-k)!⟨a,a⟩=(n-k)!\mathrm{\parallel }{a}_{\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{m}}{\mathrm{\parallel }}_{L^2}^2>0$. Distinct types are $L^2$-orthogonal, so for $x={\sum }_{p+q=k}{x}^{p,q}$ the Hermitian form $h(x,y)=Q_k(Cx,\bar{y})$ of [F1] evaluates as $h(x,x)={\sum }_{p+q=k}{i}^{p-q}{Q}_k(x^{p,q},{\bar{x}}^{p,q})={\sum }_{p+q=k}(n-k)!\mathrm{\parallel }{x}^{p,q}{\mathrm{\parallel }}^2>0$; hence $Q_k$ is nondegenerate and satisfies the polarization conditions of [F1], the rationality clause aside.

Assume now that $[\omega ]=\rho (\mathrm{\ell })$ with $\mathrm{\ell }\in H^2(X;\mathbb{Q})$. By the rationality clause of [F3] the primitive space $P^k=\ker L^{n-k+1}$ is a rational subspace of $H^k(X;\mathbb{Q})$ and its complexification is preserved by the Weil operator, and by [F8] the rational and de Rham structures are the natural ones compatible with cup product. For rational classes $a,b\in P^k$ the top-integral lemma [F5] and the compatibility of the de Rham comparison with cup product identify $Q_k(a,b)=ϵ(k){\int }_{X}a\wedge b\wedge {\omega }^{n-k}$ with $ϵ(k)⟨a⌣b⌣{\mathrm{\ell }}^{n-k},[X]⟩$, the Kronecker evaluation of a rational cohomology class against the integral fundamental class, hence a rational number; step 1.1 then exhibits $Q_k$ as a rational polarization of the pure Hodge structure $P^k$ in the sense of [F1, F4].

For a class of degree $k\le n$ write $a={\sum }_r{L}^r{a}_r$, $b={\sum }_s{L}^s{b}_s$ with $a_r\in P^{k-2r}$ and $b_s\in P^{k-2s}$ by the Lefschetz decomposition of [F3], and set $S_k(a,b):={\sum }_r(-1{)}^r{Q}_k(L^r{a}_r,L^r{b}_r)$. Since $ϵ(k)=ϵ(k-2r+2r)=(-1{)}^rϵ(k-2r)$, one has $(-1{)}^r{Q}_k(L^{r}u,L^{r}v)=Q_{k-2r}(u,v)$ for $u,v\in P^{k-2r}$: both sides are integrals of $u\wedge v\wedge {\omega }^{n-k+2r}$ with the respective constants. Hence on the summand $L^r{P}^{k-2r}$ the form $S_k$ is the transport of the polarization $Q_{k-2r}$ via $L^r$, and distinct summands are orthogonal, because for $r<s$ the integral of $L^r{a}_r\wedge L^s{b}_s\wedge {\omega }^{n-k}$ equals ${\int }_X{a}_r\wedge b_s\wedge {\omega }^{n-k+r+s}$ up to sign and $n-k+r+s\ge n-(k-2r)+1$, so the integral vanishes: on harmonic primitive representatives, $L^{n-(k-2r)+1}{a}_r=0$ by [F2], and changing to those representatives does not change the integral by step 1.1. Since $L$ shifts both Hodge indices by $1$ and commutes with the Weil operator, which acts by type scalars, the positivity of step 1.1 transfers to every summand, so $S_k$ is nondegenerate, $(-1{)}^k$-symmetric, type-orthogonal with positive definite Weil form, and rational when $\mathrm{\ell }$ is rational: a rational polarization of the pure Hodge structure $H^k$. For $n<k\le 2n$ hard Lefschetz [F3] gives the rational isomorphism $L^{k-n}:H^{2n-k}\to H^k$, which shifts both Hodge indices by $k-n$, and the transported form $S_k(a,b):=S_{2n-k}(L^{-(k-n)}a,L^{-(k-n)}b)$ is again a polarization, of the Tate twist $H^{2n-k}(-(k-n))$. For $k>2n$, [F4] computes cohomology by a de Rham complex with no degree-$k$ forms, so $H^k=0$ and its unique bilinear form is a vacuous polarization. This proves the polarization clauses; replacing the rational class by an arbitrary real Kähler class keeps the positivity and nondegeneracy arguments of steps 1.1 and 2.1 for that metric but drops only rationality.

Let $Y\subseteq {\mathbb{P}}^M$ be a smooth closed complex submanifold; the restriction of the Fubini–Study form is a Kähler form on $Y$, and its class is the real image of the integral hyperplane class $c_1(\mathcal{O}(1){\mathrm{\mid }}_Y)$ by [F7], so the hyperplane class is rational and steps 1.1-3.1 apply to $Y$. For a smooth projective complex variety $X$ with ample class $\mathrm{\ell }$, the analytification $X^{\mathrm{a}\mathrm{n}}$ is compact Kähler and [F7] supplies a positive closed real $(1,1)$-form $\eta$ representing $\rho (\mathrm{\ell })$; $\eta$ is the fundamental form of a Kähler metric on $X^{\mathrm{a}\mathrm{n}}$ (its $(1,1)$-form is positive and closed), and applying steps 1.1-3.1 to $(X^{\mathrm{a}\mathrm{n}},\eta )$ proves the smooth projective clauses. The same argument applies to every real Kähler class, since such a class is by definition represented by a Kähler form, and to $\mathrm{\ell }+ϵ\eta$ whenever that class is represented by a positive closed form; in these last cases no rationality is claimed.

Fix $0\le k\le n$ and a continuous family $t\mapsto {\mathrm{\ell }}_t$ of real Kähler classes on the fixed $X$, and put $A(t):=L(t{)}^{n-k+1}:H^k(X;\mathbb{R})\to H^{2n-k+2}(X;\mathbb{R})$, cup product with ${\mathrm{\ell }}_t^{n-k+1}$. In fixed bases of the finite-dimensional real cohomology spaces the matrix of $A(t)$ depends continuously on $t$, and its rank is the topological constant $\dim H^{k-2}(X;\mathbb{R})$: the Lefschetz decomposition of [F3] writes $H^k=P^k\oplus L{H}^{k-2}$ with $P^k=\ker A(t)$, and $L:H^{k-2}\to H^k$ is injective by hard Lefschetz for $k-2\le n$, the cases $k=0,1$ having $H^{-2}=H^{-1}=0$. At any parameter $t_0$ choose a nonvanishing maximal minor of the matrix of $A(t_0)$; by continuity it stays invertible for $t$ near $t_0$, and solving the corresponding pivot equations expresses $\ker A(t)=P^k(t)$ as the graph of a continuously varying matrix over the complementary coordinate subspace. Hence the primitive kernels of every fixed degree $0\le k\le n$ vary continuously as subspaces. For each parameter a Kähler representative exists by the definition of Kähler class; positivity at that parameter is that of step 1.1 for this representative and is not inferred by continuity at a limiting non-Kähler class, and rationality of primitive spaces is asserted only at rational classes by step 2.1.