Projective Hodge Polarization Foundations

Let $x\in X(\mathbb{C})$ be a closed point. Since $X\to \mathrm{Spec}\mathbb{C}$ is smooth and locally of finite presentation, the Jacobian criterion [F4] at $x$ gives a standard smooth presentation $C_h\cong (\mathbb{C}[t_1,\dots ,t_m]\mathrm{/}(f_1,\dots ,f_r){)}_g$ in which some $r\times r$ minor of $\left(\mathrm{\partial }{f}_j\mathrm{/}\mathrm{\partial }{t}_i\right)$ is invertible, the components of the chart having dimension $m-r$; as $X$ has pure dimension $n$, this gives $m-r=n$ [F4]. The complex points of the chart model are $Z=\{t\in {\mathbb{C}}^m:f_1(t)=\cdots =f_r(t)=0,g(t)\mathrm{\ne }0\}$, and the point $x$ corresponds to a point of $Z$. If $r=0$, then $Z=\{g\mathrm{\ne }0\}\subseteq {\mathbb{C}}^m={\mathbb{C}}^n$ is an open coordinate set and the ambient coordinates are the chart. If $r\ge 1$ and $n\ge 1$, reorder the coordinates as $t=(t^{\mathrm{\prime }},t^{\mathrm{\prime }\mathrm{\prime }})\in {\mathbb{C}}^n\times {\mathbb{C}}^r$ so that $\det \left(\mathrm{\partial }{f}_j\mathrm{/}\mathrm{\partial }{t}_k^{\mathrm{\prime }\mathrm{\prime }}\right)$ is invertible at the point; the holomorphic implicit function theorem [F5] then gives a holomorphic $\phi :A\to {\mathbb{C}}^r$ on a neighbourhood $A$ of $x^{\mathrm{\prime }}$ with $f(t^{\mathrm{\prime }},\phi (t^{\mathrm{\prime }}))=0$ and, after shrinking and imposing $g\mathrm{\ne }0$, with $f(t)=0⟺t^{\mathrm{\prime }\mathrm{\prime }}=\phi (t^{\mathrm{\prime }})$ on $A\times B$, so that $t^{\mathrm{\prime }}\mapsto (t^{\mathrm{\prime }},\phi (t^{\mathrm{\prime }}))$ is a homeomorphism of a neighbourhood of $x^{\mathrm{\prime }}$ onto a neighbourhood of $x$ in $Z$, the inverse being the projection $(t^{\mathrm{\prime }},t^{\mathrm{\prime }\mathrm{\prime }})\mapsto t^{\mathrm{\prime }}$; this is a chart of a complex manifold of dimension $n$ [F2]. If $n=0$ and $m=r\ge 1$, the inverse function theorem in several variables [F5] applied to $(f_1,\dots ,f_r):{\mathbb{C}}^m\to {\mathbb{C}}^m$, whose Jacobian is invertible at $x$, shows that $Z$ is locally the single point $x$, a chart into ${\mathbb{C}}^0=\{0\}$; these singleton charts give the discrete structure on $X(\mathbb{C})$ in the $n=0$ convention of [F1], and if $X=\mathrm{\varnothing }$ there are no closed points and we set $X^{\mathrm{a}\mathrm{n}}=\mathrm{\varnothing }$. If $X\mathrm{\ne }\mathrm{\varnothing }$, then $X(\mathbb{C})\mathrm{\ne }\mathrm{\varnothing }$: a nonempty affine open has a nonzero finite-type complex coordinate ring, hence a maximal ideal by [F11]; its residue field is finite over the algebraically closed $\mathbb{C}$, hence equals $\mathbb{C}$. Such a rational point is closed in the whole finite-type scheme: any specialization lies in an affine neighbourhood that also contains the original point, which is maximal there, so the specialization equals that point. The same residue-field argument identifies all closed points with complex-valued coordinate points.

On the standard chart $U_j\cong {\mathbb{C}}^N$ of ${\mathbb{P}}^N(\mathbb{C})$ put ${\omega }_{\mathrm{F}\mathrm{S}}:=\frac{i}{2\pi }\mathrm{\partial }\bar{\mathrm{\partial }}\log (1+\mathrm{\mid }{z}^{(j)}{\mathrm{\mid }}^2)$, $\mathrm{\mid }{z}^{(j)}{\mathrm{\mid }}^2={\sum }_l\mathrm{\mid }{z}_l^{(j)}{\mathrm{\mid }}^2$. On an overlap one has $1+\mathrm{\mid }{z}^{(j)}{\mathrm{\mid }}^2=\mathrm{\mid }h{\mathrm{\mid }}^2(1+\mathrm{\mid }{z}^{(k)}{\mathrm{\mid }}^2)$ for the nonvanishing holomorphic function $h=Z_k\mathrm{/}{Z}_j$, and $\mathrm{\partial }\bar{\mathrm{\partial }}\log \mathrm{\mid }h{\mathrm{\mid }}^2=0$ because locally one can choose a holomorphic logarithm of $h$: near a point write $h=h_0(1+v)$ with $\mathrm{\mid }v\mathrm{\mid }<1$ and use the convergent power series for $\log (1+v)$; then $\log \mathrm{\mid }h{\mathrm{\mid }}^2$ is a constant plus that holomorphic logarithm and its conjugate; hence the local forms agree on overlaps and define a global smooth real form of type $(1,1)$ on ${\mathbb{P}}^N(\mathbb{C})$ [F1]. It is closed, since $d(\mathrm{\partial }\bar{\mathrm{\partial }}f)=\bar{\mathrm{\partial }}\mathrm{\partial }\bar{\mathrm{\partial }}f=0$ for smooth $f$. It is positive: in a chart its Hermitian matrix is $((1+\mathrm{\mid }z{\mathrm{\mid }}^2){\delta }_{jk}-{\bar{z}}_j{z}_k)\mathrm{/}(2\pi (1+\mathrm{\mid }z{\mathrm{\mid }}^2{)}^2)$, and for $v\mathrm{\ne }0$ the numerator quadratic form equals $(1+\mathrm{\mid }z{\mathrm{\mid }}^2)\mathrm{\mid }v{\mathrm{\mid }}^2-\mid {\sum }_j{z}_j{v}_j{\mid }^2>0$ by Cauchy-Schwarz. For $N=0$, ${\mathbb{P}}^0(\mathbb{C})$ is a point and ${\omega }_{\mathrm{F}\mathrm{S}}=0$.

On overlaps, regular coordinate functions are locally fractions with nonvanishing denominators, hence holomorphic on the graph charts of step 1.1; the reverse transition is obtained in the same way. Zariski-open subsets are open for this topology, because on an affine presentation their complements are polynomial zero loci. This gives compatible local holomorphic atlases. For a smooth separated finite-type complex scheme $Y$, choose a finite affine cover, available by finite type; each affine point locus is a subspace of a finite-dimensional complex coordinate space and is second-countable, so the resulting finite open cover makes $Y^{\mathrm{a}\mathrm{n}}$ second-countable. Its diagonal is closed: on each product of affine opens, [F12] realizes the scheme diagonal as a closed subscheme, whose complex point locus is cut out by polynomial equations and hence is Euclidean closed. A closed diagonal implies Hausdorffness, since its open complement supplies product neighbourhoods separating any two distinct points. Thus these charts define a complex manifold for $Y$ (with the dimension-zero and empty conventions of [F1]). Applying the same construction to a smooth separated finite-type source $Z$, a morphism $f:Z\to Y$ sends complex-valued points to complex-valued points by composition with $\mathrm{Spec}\mathbb{C}\to Z$, and its local expressions are regular functions, hence holomorphic. To see continuity before using those expressions, inverse images of target affine opens are Zariski-open and therefore open, and within them every target coordinate is continuous. Thus $f^{\mathrm{a}\mathrm{n}}$ is a holomorphic map.

Let $\mathcal{O}(1)$ be the holomorphic line bundle on ${\mathbb{P}}^N(\mathbb{C})$ with transition functions $g_{jk}=Z_j\mathrm{/}{Z}_k$ on the standard cover, and give it the Hermitian metric whose local weight in the $j$-th trivialization is ${\psi }_j=(1+\mathrm{\mid }{z}^{(j)}{\mathrm{\mid }}^2{)}^{-1}>0$; on overlaps ${\psi }_j=\mathrm{\mid }{g}_{jk}{\mathrm{\mid }}^2{\psi }_k$, so the weights define a Hermitian metric [F2]. The local forms ${\omega }_j:={\psi }_j^{-1}\mathrm{\partial }{\psi }_j$ satisfy ${\omega }_j={\omega }_k+g_{jk}^{-1}d{g}_{jk}$ because $\mathrm{\partial }\log \mathrm{\mid }{g}_{jk}{\mathrm{\mid }}^2=g_{jk}^{-1}\mathrm{\partial }{g}_{jk}$ for holomorphic nonvanishing $g_{jk}$, so they glue to a complex connection $\mathrm{\nabla }$ on $\mathcal{O}(1)$ [F8]; and $\mathrm{\nabla }$ is Hermitian, because ${\omega }_j+{\bar{\omega }}_j={\psi }_j^{-1}\mathrm{\partial }{\psi }_j+{\psi }_j^{-1}\bar{\mathrm{\partial }}{\psi }_j={\psi }_j^{-1}d{\psi }_j$ is exactly the metric-compatibility identity $d{\psi }_j={\omega }_j{\psi }_j+{\psi }_j{\bar{\omega }}_j$ in the rank-one holomorphic frame [F8]. Its curvature is $\mathrm{\Omega }=d{\omega }_j+{\omega }_j\wedge {\omega }_j=d{\omega }_j$, the wedge term vanishing in rank one, and $d{\omega }_j=\bar{\mathrm{\partial }}({\psi }_j^{-1}\mathrm{\partial }{\psi }_j)=\bar{\mathrm{\partial }}\mathrm{\partial }\log {\psi }_j=-\mathrm{\partial }\bar{\mathrm{\partial }}\log {\psi }_j$ [F8]; for ${\psi }_j=(1+\mathrm{\mid }{z}^{(j)}{\mathrm{\mid }}^2{)}^{-1}$ this gives $\mathrm{\Omega }=\mathrm{\partial }\bar{\mathrm{\partial }}\log (1+\mathrm{\mid }{z}^{(j)}{\mathrm{\mid }}^2)$, so $-\mathrm{\Omega }\mathrm{/}(2\pi i)=\frac{i}{2\pi }\mathrm{\partial }\bar{\mathrm{\partial }}\log (1+\mathrm{\mid }{z}^{(j)}{\mathrm{\mid }}^2)={\omega }_{\mathrm{F}\mathrm{S}}$. The first-Chern comparison [F7] applied to $\mathcal{O}(1)$ with this Hermitian metric and Hermitian connection therefore gives $J_{{\mathbb{P}}^N(\mathbb{C})}[{\omega }_{\mathrm{F}\mathrm{S}}]=\rho (c_1(\mathcal{O}(1)))$.

By [F6] choose $d\ge 1$, $N\ge 0$ and a closed immersion $j:X\to {\mathbb{P}}_{\mathbb{C}}^N$ with $j^{\ast }\mathcal{O}(1)\cong A^{\otimes d}$. Its complex point map is injective and its image is closed in ${\mathbb{P}}^N(\mathbb{C})$: in each standard affine chart [F12] presents the image as the polynomial zero locus of a quotient ideal. The atlas topology agrees with this subspace topology. Indeed, on such an affine chart, the source coordinate algebra is a quotient of the ambient polynomial algebra by [F12]. Localizing and choosing any finite presentation of that algebra does not change its Euclidean point topology: each generator in either presentation is a polynomial or a fraction in the other generators with nonvanishing denominator, so the induced point maps are continuous inverses; localized principal opens are Euclidean open. The graph neighbourhoods of step 1.1 are precisely open subsets of these affine point loci. Hence $j^{\mathrm{a}\mathrm{n}}$ is a homeomorphism onto its closed image, and [F3] gives compactness, Hausdorffness and second-countability. Its holomorphic differential is injective: the free local coordinates $t_1^{\mathrm{\prime }},\dots ,t_n^{\mathrm{\prime }}$ of step 1.1 lift through the localized quotient map to regular functions $q_1,\dots ,q_n$ on a neighbourhood of $j(x)$ in the ambient affine chart; for the local holomorphic expression $g$ of $j$, $(q_1,\dots ,q_n)\circ g=t^{\mathrm{\prime }}$ in those coordinates, so differentiation gives a left inverse to $d{g}_x$. If $n=0$, the topology is discrete by step 1.1 and compactness now makes $X(\mathbb{C})$ finite; the empty case is immediate.

Pull the Hermitian metric and the Hermitian connection of step 2.2 back along the holomorphic map $j^{\mathrm{a}\mathrm{n}}:X^{\mathrm{a}\mathrm{n}}\to {\mathbb{P}}^N(\mathbb{C})$ to the line bundle $A^{\otimes d}$ on $X^{\mathrm{a}\mathrm{n}}$; in a pulled-back local frame its connection form is $j^{\ast }{\omega }_j$, whose gauge law is the pullback of that in step 2.2; pulling back the metric-compatibility identity proves it is Hermitian, and exterior differentiation gives curvature $d(j^{\ast }{\omega }_j)=j^{\ast }d{\omega }_j=j^{\ast }\mathrm{\Omega }$ [F8], and ${\eta }_{A^{\otimes d}}:=-j^{\ast }\mathrm{\Omega }\mathrm{/}(2\pi i)=j^{\ast }{\omega }_{\mathrm{F}\mathrm{S}}$ is a smooth closed real $(1,1)$-form, positive because the differential of $j$ is injective at every point of $X(\mathbb{C})$ by step 3.1 while ${\omega }_{\mathrm{F}\mathrm{S}}$ is positive. The first-Chern comparison [F7] applied on the compact manifold $X^{\mathrm{a}\mathrm{n}}$ to $(A^{\otimes d},j^{\ast }\mathrm{\nabla })$ gives $J_{X^{\mathrm{a}\mathrm{n}}}[{\eta }_{A^{\otimes d}}]=\rho (c_1(A^{\otimes d}))$, and the tensor-power additivity [F9] gives $c_1(A^{\otimes d})=d{c}_1(A)$, by [F13] the smooth line bundles are numerable and each connected manifold component is path-connected, paracompact Hausdorff, compactly generated weak Hausdorff and of CW homotopy type. Components are open, so the identity applies componentwise and hence globally; compactness permits only finitely many components. With the natural comparisons of [F10], ${\eta }_A:=d^{-1}{\eta }_{A^{\otimes d}}$ is again smooth, closed, real, of type $(1,1)$ and positive, and $J_{X^{\mathrm{a}\mathrm{n}}}[{\eta }_A]=d^{-1}\rho (d{c}_1(A))=\rho (c_1(A))$. For $X=\mathrm{\varnothing }$ or $n=0$ all forms of positive degree and all degree-two classes are zero, and ${\eta }_A=0$ satisfies the identity trivially. This proves the lemma for every ample $A$, the Axiom of Choice being used through the declared algebraic and comparison suppliers [F4, F6, F7, F9, F10, F11, F12, F13].

By [F3] the Laplacian commutes with $L$, $\mathrm{\Lambda }$ and $H$; hence $L$ and $\mathrm{\Lambda }$ carry harmonic forms to harmonic forms and $H$ acts on a harmonic form of degree $k$ by the scalar $k-n$ [F1]. By [F4] every real de Rham class has a unique harmonic representative, so applying the corresponding operator to that representative defines $\mathbb{R}$-linear maps $L,\mathrm{\Lambda },H$ on $H^{\ast }(X;\mathbb{R})$; on closed forms $L$ is wedge with the closed $\omega$, so on cohomology it is cup product with $[\omega ]$, and the ${\mathfrak{s}\mathfrak{l}}_2$ relations $[L,\mathrm{\Lambda }]=H$, $[H,L]=2L$, $[H,\mathrm{\Lambda }]=-2\mathrm{\Lambda }$ hold on forms by [F2], hence on harmonic forms by commutation with the Laplacian, hence pass to cohomology. Unique harmonic representatives also transport the pointwise primitive decomposition: a harmonic form $u$ of degree $k$ is uniquely ${\sum }_r{L}^r{u}_r$ with $u_r$ primitive harmonic of degree $k-2r$, $r\ge \max (0,k-n)$, because its primitive-component maps are polynomials in $L,\mathrm{\Lambda }$ by [F2], they commute with $\mathrm{\Delta }$ by [F3] and take a smooth harmonic form to smooth harmonic primitive components; conversely every sum of string elements with primitive harmonic seeds is harmonic. Since $L^{n-t+1}{u}_r=0$ and $L^r{u}_r\mathrm{\ne }0$ for $0\le r\le n-t$ when $t=\mathrm{deg}{u}_r\le n$ by [F2], the map $L^{n-k}$ sends the string of $u_r$ bijectively from its $(k)$-position to its $(2n-k)$-position and zero is the only string element killed in the relevant range, so $L^{n-k}:H^k(X;\mathbb{R})\to H^{2n-k}(X;\mathbb{R})$ is an isomorphism for $0\le k\le n$; the kernel of $L^{n-k+1}$ in degree $k\le n$ consists exactly of the sums ${\sum }_r{L}^r{u}_r$ with $r=0$, because $L^{n-k+1}{L}^r{u}_r=L^{r+n-k+1}{u}_r=0$ if and only if $n-k+1+r>n-t$ with $k=t+2r$, that is $r=0$; this is the displayed $P^k$. Summing the decompositions string by string gives $H^k(X;\mathbb{R})={⨁}_{r\ge \max (0,k-n)}{L}^r{P}^{k-2r}$, and since $L$ raises both Hodge indices by one and $\mathrm{\Lambda }$ lowers both by one, a class of type $(p,q)$ has primitive components of type $(p-r,q-r)$.

Assume $[\omega ]$ is the real image of a rational class $\mathrm{\ell }$, so that cup product with $[\omega ]$ on cohomology is the scalar extension of the rational map $a\mapsto a⌣\mathrm{\ell }$ [F4, F5]. In rational bases the matrix of $L^{n-k}:H^k(X;\mathbb{Q})\to H^{2n-k}(X;\mathbb{Q})$ is rational and becomes invertible over $\mathbb{R}$ by step 1.1, so its determinant is a nonzero rational number and the map is already invertible over $\mathbb{Q}$; under the comparison of [F5] its scalar extension to any characteristic-zero field $F$ is therefore an isomorphism $H^k(X;F)\to H^{2n-k}(X;F)$, compatible with cup products. For $0\le k\le n$, the subspaces $P^k=\ker L^{n-k+1}$ are rational; for $k>n$ they are zero by [F2]. Thus the primitive summands $L^r{P}^{k-2r}$ are then rational, and the rational matrix of the direct-sum map ${⨁}_{r\ge \max (0,k-n)}{L}^r{P}^{k-2r}\to H^k(X;\mathbb{Q})$ becomes the isomorphism of step 1.1 over $\mathbb{R}$, hence is invertible with nonzero rational determinant and stays invertible after scalar extension, which proves that the decomposition is rational and persists over $F$. Finally the lowering operator reconstructed on each string by $\mathrm{\Lambda }{L}^{r}u=r(n-t-r+1)L^{r-1}u$ for primitive $u$ of degree $t=k-2r$ [F2] has rational coefficients and, by uniqueness of the primitive decomposition over $\mathbb{R}$, it coincides there with the transported harmonic operator $\mathrm{\Lambda }$; so the ${\mathfrak{s}\mathfrak{l}}_2$ structure itself is rational.

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.