Geometric Gauss-Manin Hodge variations and integral lattices

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.