Polarized Hodge and hard Lefschetz package for smooth projective varieties

By [F1] the analytification $X^{\mathrm{a}\mathrm{n}}$ is a compact Kähler manifold of dimension $n$, and the ample class $\mathrm{\ell }$ is the real image of an integral class represented by a positive closed real $(1,1)$-form, so $\mathrm{\ell }$ is a real Kähler class and has Hodge type $(1,1)$ in the Hodge structure of [F2]. By [F2] each rational cohomology space $H^k(X;\mathbb{Q})$ is finite-dimensional and pure Hodge of weight $k$, pullback along holomorphic maps of analytifications is a morphism of Hodge structures because it preserves the type decomposition, and cup product has Hodge type $(0,0)$ since the wedge of forms of types $(p,q)$ and $(r,s)$ has type $(p+r,q+s)$; the rational structure and its scalar extensions are those of the comparison [F2].

Apply the hard-Lefschetz and primitive-decomposition theorem [F3] to $X^{\mathrm{a}\mathrm{n}}$ with the Kähler class $\mathrm{\ell }$: the maps ${\mathrm{\ell }}^{n-k}:H^k\to H^{2n-k}$ are isomorphisms for $0\le k\le n$, the primitive spaces are $P^k=\ker {\mathrm{\ell }}^{n-k+1}$ in those degrees, the decomposition $H^k={⨁}_{r\ge \max (0,k-n)}{\mathrm{\ell }}^r{P}^{k-2r}$ holds in every degree, a class of type $(p,q)$ has primitive components of type $(p-r,q-r)$, and all of this data is rational because $\mathrm{\ell }$ is rational. For $n<k\le 2n$ the complementary degree $2n-k$ satisfies $0\le 2n-k<n$, so hard Lefschetz in that degree gives that ${\mathrm{\ell }}^{n-(2n-k)}={\mathrm{\ell }}^{k-n}:H^{2n-k}\to H^k$ is an isomorphism; its inverse is likewise rational. For $k>2n$, both $H^{2n-k}$ and $H^k$ are zero: negative-degree cohomology is zero, and [F2] computes $H^k$ with the de Rham complex, which has no forms above real dimension $2n$. Thus the same displayed map is the unique isomorphism of zero spaces.

By the Hodge–Riemann theorem [F4] applied to the same compact Kähler manifold, the rational primitive summands of step 2.1 satisfy the displayed Hodge–Riemann positivity and the sign-adjusted full cohomology pairing is a rational polarization of $H^k$, with the transport from the complementary degree $\le n$ and the Tate shift in degrees $k>n$; the same absolute positivity and hard-Lefschetz statements hold for every real Kähler class, and the primitive kernels of each fixed degree vary continuously along a continuous family of Kähler classes [F4]. The complementary-degree rational cup-evaluation pairing is also nondegenerate. For $0\le k\le n$, precomposing its second variable with the isomorphism $L^{n-k}$ of step 2.1 gives $ϵ(k)Q_k$ on $H^k$; the orthogonality and signed summand description of [F4] make $Q_k$ nondegenerate, since changing a nondegenerate block by the sign $(-1{)}^r$ does not change its radical. For $k>n$ transpose the pairing in the complementary degree; outside $0\le k\le 2n$ both spaces are zero. By the orthogonal-splitting lemma [F5] every rational sub-Hodge structure of $H^k$ splits off orthogonally with polarized complement, every short exact sequence of polarizable rational Hodge structures splits, and every polarizable structure is a finite direct sum of simple ones. The rationality statements of step 2.1 and of this step are rational linear identities (nonzero rational determinants, rational idempotent projections, rational matrices of the Lefschetz isomorphisms), so by [F5] they persist under extension of scalars to every characteristic-zero field. No supplier in [F3, F4, F5] asserts or uses weak hyperplane Lefschetz, relative hard Lefschetz, singular intersection-cohomology purity or semisimplicity of arbitrary local systems, so this package supplies none of them.