Geometric Fixed Parts And Gauss Manin Semisimplicity

Identify the radical of the regular trace form with $J$. The ideal $J$ is nilpotent and $B\mathrm{/}J$ is semisimple, by [F1]. Choose a composition series of the left regular $B$-module; the simple factors are killed by $J$, since $JS=S$ for a simple factor would give $J^{k}S=S$ for every $k$, contradicting nilpotence. Each simple type of $B\mathrm{/}J$ occurs among these factors: $B\mathrm{/}J$ is a quotient of $B$, so its left regular module is a quotient of the regular module and its composition factors, namely $d_i\ge 1$ copies of each column module, occur in the series. In a basis adapted to the series the diagonal blocks of $L_b$ are the factor actions, so $\mathrm{tr}(L_b)={\sum }_i{m}_i\mathrm{tr}({\bar{b}}_i)$ with multiplicities $m_i\ge 1$. Therefore the radical $\{b:(b,c)=0\text{ for all }c\}$ of the trace form is exactly $J$: it contains $J$, and on $B\mathrm{/}J$ the induced form is the sum over $i$ of $m_i$ times the nondegenerate matrix trace pairing of the factor $M_{d_i}(\mathbb{C})$, which is nondegenerate because $m_i\mathrm{\ne }0$ and $\mathbb{C}$ has characteristic zero, by [F2].

Descend the kernel to a rational subspace. Evaluate the trace form in the rational basis $a_1,\dots ,a_d$ of $A$: the Gram matrix $T=(\mathrm{tr}(L_{a_i{a}_j}){)}_{i,j}$ has rational entries, because each $L_{a_i{a}_j}$ is the extension to $\mathbb{C}$ of the rational endomorphism of $A$ given by left multiplication by $a_i{a}_j$, and the trace of such an extension is the trace of a rational matrix. By step 1.1 its kernel in $B$ is $J$. Let $J_0=\{a\in A:Ta=0\}\subseteq A$ be the kernel computed in $\mathbb{Q}$-coordinates, a rational subspace; then $J_0\otimes \mathbb{C}=J$, since a rational linear system has the same solutions after extension of scalars and its solution space over $\mathbb{C}$ is the complexification of its rational solution space. Since $J$ is a two-sided ideal of $B$, the intersection $J_0=J\cap A$ is a two-sided ideal of $A$, and $W=\{v\in V:J_{0}v=0\}$ is a rational $A$-submodule of $V$.

Identify $W_{\mathbb{C}}$ with the socle. Because $J_0\otimes \mathbb{C}=J$, one has $W_{\mathbb{C}}=\{v\in V_{\mathbb{C}}:Jv=0\}$: choose a finite rational basis $j_1,\dots ,j_a$ of $J_0$, so $W$ is the kernel of $V\to V^a$, $v\mapsto (j_{1}v,\dots ,j_{a}v)$. Write an element of $V_{\mathbb{C}}$ as ${\sum }_{i=1}^k{v}_i\otimes {\lambda }_i$ with the finitely many ${\lambda }_i$ rationally independent, obtained by reducing the span of its original coefficients. The equations $j_a{\sum }_i{v}_i\otimes {\lambda }_i=0$ imply $j_a{v}_i=0$ for each $a,i$, by comparing coordinates in a finite rational basis of the span of the vectors $j_a{v}_i$. Thus it lies in $W\otimes \mathbb{C}$; the reverse inclusion is immediate. No basis of an infinite-dimensional space is selected. Every simple $B$-submodule $S\subseteq V_{\mathbb{C}}$ is killed by $J$: otherwise $JS=S$, and then $J^{k}S=S$ for all $k$, contradicting nilpotence of $J$. Hence the socle of $V_{\mathbb{C}}$ is contained in $\{v:Jv=0\}$. Conversely $\{v:Jv=0\}$ is a module over the semisimple algebra $B\mathrm{/}J$, and it is a sum of simple modules: for each of its vectors $v$, the cyclic module $(B\mathrm{/}J)v$ is a quotient of the regular $B\mathrm{/}J$-module, and that regular module is a finite direct sum of the column modules of [F2], so its image is a sum of simple modules. Each cyclic module is finite-dimensional, so [F3] makes its sum of simple images a finite direct sum. Every vector of $\{v:Jv=0\}$ therefore belongs to a sum of simple $B$-submodules, and this annihilator, containing every simple submodule, is exactly the socle. This argument does not require $V$ to be finite-dimensional. Thus the socle of $V_{\mathbb{C}}$ equals $W_{\mathbb{C}}=W\otimes \mathbb{C}$, a rational $A$-submodule complexified.

Nonvanishing for nonzero $V$. If $V\mathrm{\ne }0$ then $V_{\mathbb{C}}\mathrm{\ne }0$. Let $n$ be an exponent with $J^n=0$, which exists by nilpotence, and let $r\ge 1$ be least with $J^r{V}_{\mathbb{C}}=0$; such an $r$ exists since $J^n=0$. Then $J^{r-1}{V}_{\mathbb{C}}\mathrm{\ne }0$ by minimality of $r$, and $J\cdot J^{r-1}{V}_{\mathbb{C}}=J^r{V}_{\mathbb{C}}=0$, so the nonzero subspace $J^{r-1}{V}_{\mathbb{C}}$ is contained in the socle, which is therefore nonzero. By step 3.1 the socle is $W\otimes \mathbb{C}$, so $W\mathrm{\ne }0$.

Let $A\subseteq {\mathrm{End}}_{\mathbb{Q}}(V)$ be the $\mathbb{Q}$-span of the monodromy matrices. It contains the identity and is closed under multiplication, because a product of monodromy matrices is a monodromy matrix, so it is a finite-dimensional unital $\mathbb{Q}$-algebra. For every field $E$ over $\mathbb{Q}$ the canonical map $A{\otimes }_{\mathbb{Q}}E\to {\mathrm{End}}_E(V{\otimes }_{\mathbb{Q}}E)$ is injective: rationally independent matrices stay independent after extending scalars, as one sees by comparing coordinates in a rational basis of $V$. Its image is the $E$-span of the monodromy matrices, so submodules for this image are exactly the monodromy-invariant subspaces of $V{\otimes }_{\mathbb{Q}}E$; in particular the action of $A_{\mathbb{C}}=A{\otimes }_{\mathbb{Q}}\mathbb{C}$ on $V_{\mathbb{C}}$ is faithful.

Assume $V_{\mathbb{C}}$ is semisimple. Let $J=J(A_{\mathbb{C}})$. If $T\subseteq V_{\mathbb{C}}$ is a simple submodule, then $JT$ is a submodule equal to $0$ or $T$, and $JT=T$ would give $J^{k}T=T$ for every $k\ge 1$, contradicting the nilpotence of $J$ from [F1]. Hence $J$ kills every simple summand of $V_{\mathbb{C}}$ and therefore kills all of $V_{\mathbb{C}}$, since a semisimple module is a direct sum of simple submodules by [F4]. Faithfulness of the action, established in step 1.1, then forces $J=0$: an element of $J$ acts as zero on $V_{\mathbb{C}}$. Therefore $A_{\mathbb{C}}$ is semisimple.

Choose the rational basis $a_1,\dots ,a_d$ of $A$ and let $T=(\mathrm{tr}(L_{a_i{a}_j}){)}_{i,j}$ be the Gram matrix of the regular trace form of $A$ in this basis. Its entries are rational: passing from $A$ to $A_{\mathbb{C}}$ extends the rational matrix of left multiplication by $a_i{a}_j$ to a complex matrix with the same trace. By step 2.1 and [F2] the regular trace form of $A_{\mathbb{C}}$ is nondegenerate, so $\det T\mathrm{\ne }0$ as a complex number, and since $\det T\in \mathbb{Q}$ it is a nonzero rational number. For every field $k$ the scalar extension $\mathbb{Q}\hookrightarrow k$ therefore makes $T$, viewed over $k$, a nondegenerate matrix. No embedding of $k$ into $\mathbb{C}$ is used.

Let $k$ be any characteristic-zero field and put $A_k=A{\otimes }_{\mathbb{Q}}k$. For $x\in J(A_k)$ and $b\in A_k$ the product $xb$ lies in the nilpotent ideal $J(A_k)$ by [F1], so $(xb{)}^n=0$ for some $n$ and the endomorphism $L_{xb}$ of $A_k$ is nilpotent; by [F3] its trace vanishes. Nondegeneracy of the extended matrix $T$ from step 3.1 then forces $x=0$, so $J(A_k)=0$. By [F1] the quotient $A_k\mathrm{/}J(A_k)=A_k$ is a semisimple ring.

Finally $V_k=V{\otimes }_{\mathbb{Q}}k$ is semisimple over $A_k$. Choose a $\mathbb{Q}$-basis $v_1,\dots ,v_d$ of $V$. The map $(A_k{)}^d\to V_k$, $(x_1,\dots ,x_d)\mapsto {\sum }_i{x}_i{v}_i$, is a surjective $A_k$-module homomorphism, and the regular module of the semisimple ring $A_k$ is a direct sum of simple modules by [F1]. Hence $V_k$ is a sum of images of simple modules, each of which is simple or zero, and $V_k$ has finite length over $k$; by [F4] it is a finite direct sum of simple modules, that is, semisimple. By step 1.1 this is exactly semisimplicity of the monodromy module, equivalently every invariant subspace has an invariant complement by [F4], which is semisimplicity of the local system ${\mathcal{V}}_{\mathbb{Q}}\otimes k$. The zero fiber is the empty direct sum and is included.

Charts and path-connected neighbourhoods. The regular local rings of $S$ are domains by [F7]. The minimal-prime correspondence shows that only one irreducible component passes through any point; the finitely many components are therefore disjoint and both closed and open. Connectedness leaves one component. We may consequently view $S$ as an integral finite-type separated scheme by the classical-scheme equivalence in [F7]; smoothness of the algebraic variety means every local ring is regular, so by the equivalence of [F7] the structure morphism $S\to \mathrm{Spec}\mathbb{C}$ is scheme-smooth. Fix $x\in S$. Applying the Jacobian criterion of [F7] to this morphism gives an affine neighbourhood of $x$ with a 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 the Jacobian matrix $(\mathrm{\partial }{f}_j\mathrm{/}\mathrm{\partial }{t}_i)$ is invertible at $x$, and whose fibres have dimension $m-r$; here the fibre is the local variety itself, so its chart dimension is $m-r$, and the nonempty-open dimension equality of [F7] gives $m-r=n=\dim S$. If $r=0$ the chart is already open in affine space. For $r>0$, after relabelling the coordinates and applying the holomorphic implicit function theorem of [F7] (adding an unused parameter if the free-coordinate block is empty) to the holomorphic map $(f_1,\dots ,f_r)$ near the image of $x$, the common zero locus is locally the graph of a holomorphic function of the remaining $m-r=n$ coordinates; the graph coordinate map identifies a neighbourhood of $x$ in the analytic topology with an open ball in ${\mathbb{C}}^n$, and this analytic topology is the one of [F7]. The same presentation and implicit-function argument applied to the reduced induced structure of a locally closed smooth complex subvariety exhibits its points as complex charts of dimension equal to its complex dimension. Consequently $S$, and every smooth locally closed complex subvariety of $S$, is a complex manifold for the analytic topology, hence in particular a smooth real manifold of twice its complex dimension; the chart balls are path-connected neighbourhoods, so $S$ and its open subsets are locally path connected.

Smoothing a based loop. Let $\gamma :S^1\to S$ be a continuous based loop. Choose a continuous surjection $\psi :S^1\to S^1$ with $\psi (\ast )=\ast$ that is constant on a neighbourhood of $\ast$: in angular coordinates it is induced by a nondecreasing surjection $\lambda :[0,1]\to [0,1]$ constant at $0$ and $1$ on endpoint collars. The interpolation $(1-u)t+u\lambda (t)$ descends to a homotopy from the identity fixing $\ast$; then ${\gamma }_1=\gamma \circ \psi$ is a continuous based loop, constant on an open arc $W_0\ni \ast$ where ${\gamma }_1=s$, and it is based-homotopic to $\gamma$. The map ${\gamma }_1$ is smooth (indeed constant) on an open neighbourhood of the closed set $A=\{\ast \}$, so the relative Whitney approximation of [F5] applies: there are a smooth based loop ${\gamma }_2:S^1\to S$ and an open neighbourhood $W$ of $\ast$ such that ${\gamma }_2={\gamma }_1$ on $W$ and ${\gamma }_1$ is homotopic to ${\gamma }_2$ by a homotopy fixed on $W$. Thus ${\gamma }_2$ is a smooth based loop, equal to ${\gamma }_1=s$ on the open neighbourhood $W\cap W_0$ of the base point, with $[{\gamma }_2]=[{\gamma }_1]=[\gamma ]$ in ${\pi }_1(S,s)$.

A finite decomposition of the complement into codimension-two smooth pieces. The complement $Z=S\setminus U$ is closed and proper, and $S$ is irreducible by step 1.1. We construct pairwise disjoint smooth locally closed complex subvarieties $(Z_j)$ covering $Z$, each of complex dimension at most $n-1$, by downward induction on dimension. Every irreducible component $C$ of $Z$ is a proper closed irreducible subvariety of the irreducible $S$, so $\dim C\le n-1$: append $S$ to any chain of irreducible closed subsets of $C$ to obtain a chain one term longer in $S$, using the chain-dimension definition in [F7]. The same argument gives the dimension drop for every proper closed subset of an irreducible component below. Suppose a closed algebraic subset $T\subseteq Z$ is given in which every irreducible component has dimension at most $d$. For each irreducible component $C$ of $T$ of dimension exactly $d$, the regular locus $C_{\mathrm{r}\mathrm{e}\mathrm{g}}$ is a nonempty Zariski open subset of $C$: the generic-smoothness corollary of [F7], applied to the constant morphism from the irreducible classical variety $C$ to a point, produces a nonempty open subset of the regular locus, which is open by [F7]. The sets $C_{\mathrm{r}\mathrm{e}\mathrm{g}}\setminus {\bigcup }_{C^{\mathrm{\prime }}\mathrm{\ne }C}{C}^{\mathrm{\prime }}$, for the finitely many top-dimensional components $C$, are pairwise disjoint smooth locally closed complex subvarieties of dimension $d$. What they do not cover of $T$ is the union of the lower-dimensional components, the singular loci of the top-dimensional components, and the intersections of a top-dimensional component with another component; the lower-dimensional components have dimension at most $d-1$ by assumption and the other sets are proper closed subsets of top-dimensional irreducible components, hence have dimension at most $d-1$ by the chain argument, so the leftover $T^{\mathrm{\prime }}$ is a closed algebraic subset all of whose components have dimension at most $d-1$. Iterating from $T=Z$ and $d=n-1$ produces finitely many disjoint smooth locally closed pieces, and the iteration stops at dimension $0$, where the pieces are points and nothing is left over. Each piece has complex dimension at most $n-1$, so its real dimension is at most $2n-2$ and its real codimension in the smooth real manifold $S$ of step 1.1 is at least two; being a smooth locally closed complex subvariety, each piece is an embedded submanifold of $S$.

A cutoff vanishing at the base point and equal to one over the bad set. Let ${\gamma }_2$ be the smooth loop of step 1.2, denoting by ${\gamma }_2^{-1}(Z)\subseteq S^1$ the compact bad set. If it is empty, then ${\gamma }_2$ already lies in $U$ and its class is represented in $U$; assume it is nonempty. Since ${\gamma }_2$ equals the constant $s\in U$ on the open neighbourhood $W_2=W\cap W_0$ of $\ast$, the compact set $K={\gamma }_2^{-1}(Z)$ is disjoint from $W\cap W_0$. Choose an open arc $W^{\mathrm{\prime }}\ni \ast$ with $\overline{W^{\mathrm{\prime }}}\subseteq W_2$, put $U_1=S^1\setminus \overline{W^{\mathrm{\prime }}}$ and $U_2=W_2$; these open sets cover $S^1$, and $K\subseteq S^1\setminus W_2\subseteq U_1$. By the partition-of-unity theorem [F4] applied to the cover $\{U_1,U_2\}$ choose a subordinate smooth partition of unity $({\chi }_1,{\chi }_2)$ and set $\chi ={\chi }_1$. Then $\chi$ is smooth with values in $[0,1]$, equal to $1$ on the open set $S^1\setminus \mathrm{supp}({\chi }_2)\supseteq S^1\setminus W_2\supseteq K$, and equal to $0$ on the open set $S^1\setminus \mathrm{supp}({\chi }_1)\supseteq S^1\setminus U_1=\overline{W^{\mathrm{\prime }}}\ni \ast$; moreover whenever $\chi (p)=0$ we have ${\chi }_2(p)=1$, so $p\in \mathrm{supp}({\chi }_2)\subseteq W_2$ and hence ${\gamma }_2(p)=s\in U$. In particular $\chi (\ast )=0$.

The truncated perturbation family is transverse to every piece. Apply the perturbation lemma [F3] to the smooth loop ${\gamma }_2:S^1\to S$: there are an open ball $B\subseteq {\mathbb{R}}^m$, positive-dimensional as recorded in [F3] because $S\mathrm{\ne }\mathrm{\varnothing }$, and a smooth family $\mathcal{F}:S^1\times B\to S$ with ${\mathcal{F}}_0={\gamma }_2$ and with every parameter map $a\mapsto \mathcal{F}(p,a)$ a submersion; in particular the evaluation map $\mathcal{F}$ is a submersion, hence transverse to every embedded submanifold by [F2]. Define $\tilde{\mathcal{F}}:S^1\times B\to S$ by $\tilde{\mathcal{F}}(p,a)=\mathcal{F}(p,\chi (p)a)$; since the ball $B$ is centred at $0$ and $0\le \chi (p)\le 1$ this is well defined and smooth, and ${\tilde{\mathcal{F}}}_0={\mathcal{F}}_0={\gamma }_2$. For every $a\in B$ one has $\tilde{\mathcal{F}}(\ast ,a)=\mathcal{F}(\ast ,\chi (\ast )a)=\mathcal{F}(\ast ,0)={\gamma }_2(\ast )=s$, so every slice ${\tilde{\mathcal{F}}}_a$ is a based loop. At a point $(p,a)$ with $\chi (p)>0$ the derivative of $\tilde{\mathcal{F}}$ in the parameter direction is $\chi (p)$ times the parameter derivative of $\mathcal{F}$ at $(p,\chi (p)a)$, which is surjective, so $\tilde{\mathcal{F}}$ is a submersion there. Hence $\tilde{\mathcal{F}}⋔Z_j$ for every piece $Z_j$ of step 2.1: if $\tilde{\mathcal{F}}(p,a)\in Z_j\subseteq Z$ then $\chi (p)>0$, since $\chi (p)=0$ would give $\tilde{\mathcal{F}}(p,a)=\mathcal{F}(p,0)={\gamma }_2(p)$ with ${\gamma }_2(p)\in U$ by step 2.2, and at a point with $\chi (p)>0$ the map is a submersion and therefore transverse by [F2].

A parameter outside all bad sets, and surjectivity. For each piece $Z_j$ of step 2.1 the set $N_j\subseteq B$ of parameters $a$ with ${\tilde{\mathcal{F}}}_a$ not transverse to $Z_j$ is null by the parametric transversality theorem [F2]. Since there are finitely many pieces, the union $N={\bigcup }_j{N}_j$ is null in the positive-dimensional smooth manifold $B$ by [F6], so its complement is dense by [F6] and in particular nonempty; choose $a\in B\setminus N$. Then the slice ${\tilde{\mathcal{F}}}_a:S^1\to S$ is a smooth based loop transverse to every $Z_j$. A transverse map from a one-dimensional manifold cannot meet a piece of real codimension at least two: if ${\tilde{\mathcal{F}}}_a(p)\in Z_j$, transversality would give $\dim T_{{\tilde{\mathcal{F}}}_a(p)}S\le \dim \mathrm{im}(d({\tilde{\mathcal{F}}}_a{)}_p)+\dim T_{{\tilde{\mathcal{F}}}_a(p)}{Z}_j\le 1+(2n-2)<2n$, a contradiction. Hence ${\tilde{\mathcal{F}}}_a(S^1)$ meets no $Z_j$ and therefore lies in $U$. The map $(p,u)\mapsto \tilde{\mathcal{F}}(p,ua)$, $0\le u\le 1$, is a homotopy from ${\gamma }_2$ to ${\tilde{\mathcal{F}}}_a$ through based loops, so $[{\tilde{\mathcal{F}}}_a]=[{\gamma }_2]=[\gamma ]$ in ${\pi }_1(S,s)$ by [F8], while $[{\tilde{\mathcal{F}}}_a]$ lies in the image of ${\pi }_1(U,s)\to {\pi }_1(S,s)$ by [F8]. In the discarded case $K=\mathrm{\varnothing }$ of step 2.2 the loop ${\gamma }_2$ itself already lies in $U$, so the same conclusion holds. Since every class in ${\pi }_1(S,s)$ has a continuous based loop representative, the map ${\pi }_1(U,s)\to {\pi }_1(S,s)$ is surjective.

Path connectedness. Let $x,y\in U$. The chart balls of step 1.1 make $S$ locally path connected, so connectedness and [F9] give a continuous path $\beta$ from $x$ to $y$; reparametrize it to be constant near its endpoints. Traverse $\beta$ on the first quarter of $S^1$, its reverse on the second quarter, and the constant path at $x$ on the second half. This gives a continuous circle map constant near all junctions. Apply [F5] to this circle map, relative to the finite closed set of junctions, to obtain a smooth loop equal to the original on neighbourhoods of those junctions. Its bad set is disjoint from those neighbourhoods. Repeat step 2.2 with $W_2$ the union of these constant neighbourhoods and $W^{\mathrm{\prime }}$ a smaller union with closure in $W_2$. The same two-set partition supplies a cutoff vanishing near all junctions, equal to one near the bad set, and with every zero lying where the loop is in $U$. Steps 3.1 and 4.1 then give a perturbed loop in $U$ agreeing with the original near the junctions. Its first quarter joins $x$ to $y$ in $U$. Thus $U$ is path connected; when $Z=\mathrm{\varnothing }$ the unperturbed path already suffices.

Analytic submersion and properness. The relative projective embedding is a closed immersion $X\hookrightarrow S\times {\mathbb{P}}^N$. Its analytification is closed because the defining polynomial equations are continuous. For each compact $K\subset S^{\mathrm{a}\mathrm{n}}$, the inverse image $f^{-1}(K)$ is a closed subset of $K\times {\mathbb{P}}^N(\mathbb{C})$ and is compact: projective space is the continuous image of the unit sphere in ${\mathbb{C}}^{N+1}$, which is compact by [F12]. The finite-product assertion in [F12] makes $K\times {\mathbb{P}}^N$ compact. Thus $f^{\mathrm{a}\mathrm{n}}$ is topologically proper, in addition to algebraic properness from [F1]. The smooth Jacobian presentation and holomorphic implicit function theorem [F2] give local holomorphic submersion coordinates (with no equations there is nothing to solve, and an unused parameter handles a zero-dimensional free block). The smooth quasiprojective source and base are Hausdorff second-countable complex manifolds; they are metrizable as locally closed subspaces of projective space. Each fibre is a compact smooth projective complex manifold, possibly empty or with components of different dimensions, and [F1] supplies a Kähler form on each component.

Integral sheaf comparison. We supply the coefficient extension that [F4] states only for fields. On a metric space $M$ with a contractible-open basis let ${\mathcal{C}}^q$ be the sheafification of integer-valued singular cochains. The shrinking argument of [F11] makes every section over an open set $V$ the germ of a global cochain: choose a locally finite refinement $W_i$ with closure in a representative neighbourhood, assign to each simplex contained in some $W_i$ one representative value, and assign zero otherwise. At any point only finitely many $W_i$ can intervene; their representative germs agree on a smaller neighbourhood, so this assignment has the required germ. Consequently every ${\mathcal{C}}^q$ is flasque. The stalk complex of ${\mathbb{Z}}_M\to {\mathcal{C}}^{\bullet }$ is exact: stalk preservation, exact filtered colimits and contraction of cofinal contractible neighbourhoods in [F11] give $\mathbb{Z}$ in degree zero and zero in higher degrees. These flasque terms are acyclic for abelian sheaf cohomology: injectives are flasque by the extension-by-zero adjunction, and embedding a flasque sheaf into an injective leaves a flasque quotient with surjective section map, by [F11]; dimension shifting in the derived long exact sequence proves positive acyclicity. The acyclic-resolution theorem therefore applies. Finally the kernel of $C^{\ast }(M;\mathbb{Z})\to \mathrm{\Gamma }(M,{\mathcal{C}}^{\bullet })$ is the directed union of cochains vanishing on the small simplices of a cover. For each cover, the quotient of all integer chains by small chains is a complex of free abelian groups, since small simplices are a subset of the basis; it is acyclic by the small-chain theorem [F11]. Its boundaries are free by the PID submodule theorem [F11], so its short exact cycle-boundary sequences split by lifting bases, making it contractible. Its integer dual is thus acyclic. Directed union and the cochain long exact sequence give the natural comparison $H^q(M,{\mathbb{Z}}_M)\cong H_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}^q(M;\mathbb{Z})$. The canonical augmentation and sheafification maps commute with pullback, so the comparison is natural and applies also to open subspaces.

Transport and the cohomology local systems. Choose a smooth horizontal complement to $\ker d{f}^{\mathrm{a}\mathrm{n}}$ on $X^{\mathrm{a}\mathrm{n}}$ (locally the product charts of step 1.1 provide one; glue the local choices with a smooth partition of unity on the manifold $X^{\mathrm{a}\mathrm{n}}$ [F9]). For a smooth path $\gamma :[0,1]\to S$ and a point $x\in f^{-1}(\gamma (0))$, the horizontal ODE $\dot{x}(t)\in H_{x(t)}$ with $f(x(t))=\gamma (t)$ has a unique maximal solution by [F3]; because $f$ is proper, the lifted solution stays in the compact set $f^{-1}(K)$ for a compact neighbourhood $K$ of $\gamma ([0,1])$, so it exists on all of $[0,1]$: local ODE existence neighbourhoods cover the compact time-state set $[0,1]\times f^{-1}(K)$, and a finite subcover gives a common positive continuation interval; a finite maximal endpoint would therefore admit continuation, contradicting maximality. The resulting transport map ${\tau }_{\gamma }:f^{-1}(\gamma (0))\to f^{-1}(\gamma (1))$ is a diffeomorphism, smooth in the initial point by the smooth dependence on initial data in [F3], and an endpoint-fixed smooth homotopy of paths gives a homotopy of terminal diffeomorphisms by the parameter dependence in [F3]. The terminal point maps need not be identical, but their induced maps on cohomology are identical by homotopy invariance. Smooth representatives and homotopies suffice for the fundamental groupoid by [F10]: reparametrize paths to be constant near endpoints and extend them constantly to $\mathbb{R}$, then apply relative Whitney approximation fixing those collars. For a homotopy between smooth representatives, reparametrize both variables to have constant collars, extend to ${\mathbb{R}}^2$ by clamping the variables, and approximate relative to the boundary collars, where the map is smooth. This produces a smooth homotopy with the prescribed endpoints and boundary paths. Concatenation and reversal give composition and inverses of the cohomology transports. The assignments $x\mapsto H^m(f^{-1}(x);R)$ for $R=\mathbb{Z},\mathbb{Q}$, together with the covariant isomorphisms $T_{\gamma }=({\tau }_{\gamma }^{-1}{)}^{\ast }:H^m(f^{-1}(\gamma (0));R)\to H^m(f^{-1}(\gamma (1));R)$, therefore define functors on the fundamental groupoid of $S$, that is, $R$-module local systems in the sense of [F8]; this is the local system $R^m{f}_{\ast }^{\mathrm{a}\mathrm{n}}\underset{‾}{R}$ of the family.

Cohomology sheaves and lattices. The horizontal transport of step 2.1 along radial paths in a small convex base chart $B$ gives a smooth trivialization $f^{-1}(B)\cong B\times F$, with inverse furnished by reversed paths and smooth dependence from [F3]. For $R=\mathbb{Z},\mathbb{Q}$, the stalk of $R^m{f}_{\ast }\underset{‾}{R}$ is ${\mathrm{colim}}_{B\ni s}{H}^m(f^{-1}(B),\underset{‾}{R})$: a flasque resolution computes direct image because its restriction to every inverse image of an open set is section-acyclic, as in step 1.2, and exact stalk colimits commute with cohomology. The comparisons of [F4] and step 1.2 identify these groups with singular cohomology. Contracting $B$ gives $H^m(B\times F;R)\cong H^m(F;R)$, compatibly with transport. Thus both higher direct images are the local systems of step 2.1. Finite CW type and cellular cochains [F4] make the fibre's integral cohomology finitely generated. The finite cellular cochain complex tensored with $\mathbb{Q}$ is the rational cellular cochain complex, and rational linear-system reduction gives $H^m(F;\mathbb{Z})\otimes \mathbb{Q}\cong H^m(F;\mathbb{Q})$. By [F11], quotienting integral cohomology by torsion gives a finite free full lattice; transport preserves torsion and hence this lattice.

Continuous Hodge decomposition. Use the fixed-fibre trivialization of step 3.1. An empty fibre contributes zero; its zero-dimensional components are finite sets with constant type $(0,0)$ and require no elliptic argument. Work separately on its positive-dimensional components for the following construction. The relative projective embedding restricts the Fubini–Study form to a smooth family of fibrewise Kähler metrics. On the fixed compact $F$ let ${\mathrm{\Delta }}_t:H^2\to L^2$ be the corresponding Laplacians, using one reference metric for the Sobolev norms. Their coefficients vary continuously, so ${\mathrm{\Delta }}_t-{\mathrm{\Delta }}_0$ tends to zero in operator norm $H^2\to L^2$. Let ${\mathrm{\Pi }}_0$ and $G_0$ be the harmonic projection and Green operator from [F6]; ${\mathrm{\Delta }}_0+{\mathrm{\Pi }}_0$ is invertible with inverse $G_0+{\mathrm{\Pi }}_0$. The Neumann-series assertion of [F12] therefore gives continuous inverses $A_t^{-1}=({\mathrm{\Delta }}_t+{\mathrm{\Pi }}_0{)}^{-1}$ for nearby $t$. On $\ker {\mathrm{\Delta }}_t$, the map ${\mathrm{\Pi }}_0$ is injective, because $({\mathrm{\Delta }}_t+{\mathrm{\Pi }}_0)u=0$ would force $u=0$; by [F6] both harmonic kernels have the same dimension, the fixed Betti number, so this map is bijective. Thus $A_t^{-1}{\mathrm{\mid }}_{\ker {\mathrm{\Delta }}_0}$ is its inverse and identifies the kernels continuously. The same argument in $H^{k+2}\to H^k$, for every $k\ge 0$, gives continuity in each Sobolev norm. The continuously varying type projections on forms commute with the Kähler Laplacians by [F6] (the type-preservation calculation in its Hodge-decomposition proof, step 2.1); restricted to these identified finite-dimensional harmonic spaces they are continuous idempotents, whose ranks are locally constant and whose images vary continuously. They give the subspaces $H_t^{p,q}$ under the harmonic/de Rham identification. The pure decomposition, conjugation symmetry and metric independence in [F6] consequently give the required continuous pure Hodge decomposition of weight $m$.

Flat Lefschetz operators and polarization. Let $L$ be a relatively ample line bundle on $X$. Its first Chern class lies in $H^2(X;\mathbb{Q})$. Apply the positive Chern-metric construction of [F1] to the ample restriction on each compact fibre, extend that metric to a neighbourhood using the smooth trivialization of step 2.1, and shrink the base chart so vertical positivity persists by compactness. Relative to a fixed smooth reference metric on $L$, patch the logarithms of the local metric ratios with a smooth partition of unity pulled back from the base. On a fibre these base weights are constant, so the resulting Chern curvature is their positive convex combination; it is globally closed, represents $c_1(L)$ and restricts on every fibre to a Kähler form with rational class. No positivity in base directions is asserted. Then in the product chart the restrictions to the fibres induce one and the same class on the fixed model $F$ under the identifications ${\tau }_{\gamma }$ of step 2.1, because those identifications are induced by the homotopic family of inclusions of the fibres into $X$ and cohomology is homotopy invariant; explicitly, if ${\psi }_t:F\to f^{-1}(t)$ are the product-chart identifications, then the maps $(t,x)\mapsto {\psi }_t(x)$ exhibit the composites $i_t\circ {\psi }_t$ as homotopic to $i_s\circ {\psi }_s$, so ${\psi }_t^{\ast }(c_1(L){\mathrm{\mid }}_{f^{-1}(t)})={\psi }_s^{\ast }(c_1(L){\mathrm{\mid }}_{f^{-1}(s)})$. Cup product and integration are compatible with the transport isomorphisms, so the map $a\mapsto c_1(L)⌣a$ is a flat Lefschetz operator, and the polarized Lefschetz decomposition supplied by [F7] is flat. The forms $Q_k$ of [F7] applied to the primitive pieces give a flat rational bilinear form, since their defining integrals involve only cup products, the flat powers of the Kähler class just described, and the constant sign $ϵ(k)$; the positivity clause of [F7] holds on every fibre, and the full polarization is the sign-adjusted orthogonal sum along the flat Lefschetz decomposition, hence also flat, rational and positive. This supplies the flat rational polarization required in the definition.

Geometric closures and full lattices. All assertions are made over a fixed open piece of the base, and may be pulled back along a base change to any smooth $S^{\mathrm{\prime }}$ by change of the parameter. Products: for smooth projective $f_i:X_i\to S$, $i=1,2$, the product family $X_1{\times }_S{X}_2\to S$ is smooth projective, and the cohomological Künneth isomorphism of [F8] identifies $R^m(f_1{\times }_S{f}_2{)}_{\ast }\mathbb{Q}$ with the direct sum of tensor products $R^p{f}_{1\ast }\mathbb{Q}\otimes R^q{f}_{2\ast }\mathbb{Q}$, $p+q=m$, compatible with Hodge type because pullback and wedge of $(p,q)$ forms preserve bidegree by [F6]; this rational comparison uses [F8] over the field $\mathbb{Q}$, where the finite-dimensional homology groups are free. The ambient product-family cohomology has a full stable integral lattice by step 3.1; intersecting it with each rational tensor summand gives a full monodromy-stable lattice in that summand, since a rational basis can be multiplied by common denominators into the ambient lattice. Thus products and their full lattices are geometric without assuming that the integral cohomology of every smooth projective fibre is torsion-free. Direct sums in the same cohomological degree are realized by disjoint unions of families over $S$; different degrees are combined by the finite-sum realization clause of the definition. The degree-two cohomology of the constant family ${\mathbb{P}}^1\times S\to S$ is $\mathbb{Q}(-1)$, of type $(1,1)$, and its rational Hodge dual is $\mathbb{Q}(1)$, of type $(-1,-1)$, by the dual and Tate conventions in the definition. For a realization $H{\mathrm{\mid }}_U\subseteq {⨁}_i{R}^{m_i}(f_i{)}_{\ast }\mathbb{Q}(r_i)$, twisting by any integer $r$ gives the realization inside ${⨁}_i{R}^{m_i}(f_i{)}_{\ast }\mathbb{Q}(r_i+r)$, which is allowed directly by that definition; the same rational idempotent remains flat and preserves the shifted Hodge types. Thus all integer Tate twists have the stated geometric realizations, without identifying a positive twist with untwisted degree-two cohomology. For a pure variation of weight $w$, its flat rational nondegenerate polarization of step 5.1 gives $H^{\ast }\cong H(w)$ as Hodge variations, so the just-proved Tate realization also realizes the dual; the dual of a full stable lattice is again full and stable. Finite sums of pure weights are handled componentwise. Exterior powers are Hodge direct summands of tensor powers cut out by the rational alternating idempotent, which is flat and a Hodge morphism on every fibre; more generally a prescribed flat rational idempotent that is a Hodge morphism on every fibre cuts out a geometric Hodge direct summand of a realized variation. Intersecting such a rational direct summand with the full ambient lattice of step 3.1 gives its full monodromy-stable lattice: a rational basis of the summand can be cleared to integral vectors of the ambient lattice, and flatness of the summand makes the intersection monodromy-stable. Finally, for a morphism $g:X^{\mathrm{\prime }}\to X$ of smooth projective complex varieties, $g^{\ast }:H^m(X;\mathbb{Q})\to H^m(X^{\mathrm{\prime }};\mathbb{Q})$ preserves Hodge types by the pullback functoriality clause of [F6], so rational pullbacks between smooth projective fibres are morphisms of pure Hodge structures. This proves every closure and lattice clause of the statement concerning a single smooth projective family; the Gauss-Manin family itself is geometric and polarizable by steps 2.1-5.1.

Extension to the whole geometric category. For a geometric object on connected $S$, choose its realization open $U$. Steps 3.1–6.1 supply a full stable lattice and, in each pure weight, a flat rational polarization on $H{\mathrm{\mid }}_U$. By [F10], ${\pi }_1(U,s)\to {\pi }_1(S,s)$ is onto, so the same lattice in $H_s$ and the same bilinear form are preserved by every monodromy operator on $S$; transport extends them to $S$. The polarization's Hodge orthogonality extends from $U$ by continuity, since a proper algebraic subset has no analytic interior: on an irreducible smooth chart a regular defining function is holomorphic, and vanishing on a ball forces it to vanish identically by its power series and continuation through overlapping charts. Thus $U$ is analytically dense. The extended bilinear form is nondegenerate, being flat, and its Weil-operator Hermitian form is continuous and nondegenerate everywhere. It is positive on $U$; its eigenvalues cannot cross zero, so it is positive on all of $S$. Restrictions to Hodge direct summands are nondegenerate by positivity and have the intersected full stable lattices. Applying this argument componentwise to finite sums of pure weights completes the closure, dual, lattice and polarization assertions for all geometric objects, not just the realization open. In particular steps 2.1–5.1 construct the stated Gauss–Manin variation and its integral lattice on every nonempty connected base component. Disconnected bases are treated componentwise and the assertions over an empty base are vacuous.