Polarized rational Hodge structures split orthogonally

The Weil operator $C$ preserves $U_{\mathbb{C}}$, because $U_{\mathbb{C}}={⨁}_{p+q=k}(U_{\mathbb{C}}\cap V^{p,q})$ and $C$ multiplies each $U_{\mathbb{C}}\cap V^{p,q}$ by the scalar $i^{p-q}$; complex conjugation preserves $U_{\mathbb{C}}$ because it exchanges $V^{p,q}$ with $V^{q,p}$ and preserves $U_{\mathbb{C}}$ [F1]. The positive definite Hermitian form $h(x,y)=Q(Cx,\bar{y})$ of [F1] therefore restricts to $U_{\mathbb{C}}$ as a positive definite form, while $Q{\mathrm{\mid }}_U$ inherits rationality, the $(-1{)}^k$-symmetry and type orthogonality from $Q$. The identity $Q(Cx,y)=Q(x,C^{-1}y)$ holds on $V_{\mathbb{C}}$: both sides equal ${\sum }_{p+q=k}{i}^{p-q}Q(x^{p,q},y^{q,p})$, because $Q(x^{r,s},y^{a,b})$ vanishes unless $(a,b)=(s,r)$. If $u\in U$ lies in the radical of $Q{\mathrm{\mid }}_U$, so that $Q(u,U)=0$, then $Q(Cu,U_{\mathbb{C}})=Q(u,C^{-1}{U}_{\mathbb{C}})=0$; as $C^{-1}$ preserves $U_{\mathbb{C}}$, this says $Cu$ is again in the radical and $h(u,u)=Q(Cu,\bar{u})=0$, whence $u=0$ by positive definiteness. Therefore $Q{\mathrm{\mid }}_U$ is nondegenerate and is a polarization of $U$.

Since $Q{\mathrm{\mid }}_U$ is nondegenerate by step 1.1, the rational equations $Q(x,U)=0$ define a rational subspace $U^{{\perp }_Q}$ whose dimension is $\dim V-\dim U$, the kernel of the surjective pairing map $V\to U^{\vee }$, $x\mapsto Q(x,\cdot ){\mathrm{\mid }}_U$, having that dimension; and $U\cap U^{{\perp }_Q}=0$ because $Q{\mathrm{\mid }}_U$ is nondegenerate. Hence $V=U\oplus U^{{\perp }_Q}$ as $\mathbb{Q}$-vector spaces. Let $x={\sum }_{p+q=k}{x}^{p,q}\in U_{\mathbb{C}}^{{\perp }_Q}$ with $x^{p,q}\in V^{p,q}$, and fix $(r,s)$; for every $v\in U^{s,r}$ one has $0=Q(x,v)={\sum }_{p+q=k}Q(x^{p,q},v)=Q(x^{r,s},v),$ because $Q$ pairs $V^{p,q}$ only with $V^{q,p}$ [F1]. As $v$ ranges over $U^{s,r}=U_{\mathbb{C}}\cap V^{s,r}$ and $Q(x^{r,s},\cdot )$ is supported on $V^{s,r}$, this gives $Q(x^{r,s},U_{\mathbb{C}})=0$, so every $x^{r,s}$ lies in $U_{\mathbb{C}}^{{\perp }_Q}$ and $U^{{\perp }_Q}$ is a rational sub-Hodge structure. The Hermitian form $h$ restricts positively to $U_{\mathbb{C}}^{{\perp }_Q}$, so the argument of step 1.1 with $U$ replaced by $U^{{\perp }_Q}$ shows that $Q{\mathrm{\mid }}_{U^{{\perp }_Q}}$ is a polarization of $U^{{\perp }_Q}$. This proves the orthogonal Hodge direct sum.

Let $0\to U\stackrel{i}{\to }V\stackrel{\pi }{\to }W\to 0$ be a short exact sequence of polarizable rational pure Hodge structures of the fixed weight $k$ [F2], so that in particular $V$ is polarizable; fix a polarization $Q$ of $V$ and identify $U$ with its image under $i$, a rational sub-Hodge structure by [F2]. By step 2.1, $V=U\oplus U^{{\perp }_Q}$ and $Q{\mathrm{\mid }}_{U^{{\perp }_Q}}$ is a polarization. The restriction $\pi {\mathrm{\mid }}_{U^{{\perp }_Q}}$ is a morphism of Hodge structures, is injective because $U^{{\perp }_Q}\cap \ker \pi =U^{{\perp }_Q}\cap U=0$, and has source and target of the same finite dimension $\dim V-\dim U=\dim W$ by exactness, so it is bijective. A bijective morphism has bidegree-preserving inverse by [F2]: if $w\in W^{p,q}$ and ${\pi }^{-1}(w)=\sum x^{r,s}$ with $x^{r,s}\in V^{r,s}$, then $w=\sum \pi (x^{r,s})$ with $\pi (x^{r,s})\in W^{r,s}$, directness of the target decomposition forces $\pi (x^{r,s})=0$ for $(r,s)\mathrm{\ne }(p,q)$, and injectivity gives $x^{r,s}=0$ for those $(r,s)$. Hence $s:=(\pi {\mathrm{\mid }}_{U^{{\perp }_Q}}{)}^{-1}:W\to V$ is a morphism with $\pi \circ s={\mathrm{id}}_W$, and the sequence splits.

For semisimplicity, suppose $V\mathrm{\ne }0$ is polarizable. Among the nonzero rational sub-Hodge structures of $V$ the dimensions form a nonempty subset of $\{1,\dots ,\dim V\}$, so there is one, $U$, of least dimension; then $U$ is simple, since a nonzero rational sub-Hodge structure of $U$ is one of $V$ of no larger dimension. By step 2.1, $V=U\oplus U^{{\perp }_Q}$ with $U^{{\perp }_Q}$ polarizable of strictly smaller dimension, and induction on $\dim V$ (the case $U^{{\perp }_Q}=0$ terminating) exhibits $V$ as a finite direct sum of simple Hodge structures. A splitting produced in step 3.1 and a decomposition produced here are given by rational matrices in rational bases; the identities $\pi \circ s={\mathrm{id}}_W$, idempotence of the projections, and $Q(x,U)=0$ for $x\in U^{{\perp }_Q}$ are linear equations with rational coefficients and therefore remain valid after extension of scalars to any characteristic-zero field $F$, so the extended sequence splits and the extended decomposition is a direct sum decomposition of $V_F$ over $F$. Positivity is used only over $\mathbb{R}$ and is not invoked over $F$, and no new subobject of $V_F$ is claimed to descend to $V$. The argument is finite-dimensional rational and complex linear algebra on fixed finite-dimensional structures and uses no choice principle.