The fixed part of a geometric Hodge variation

Leray edge map in the family case. Let $f:X\to S$ be smooth projective and put $H=R^m{f}_{\ast }\mathbb{Q}$. By [F3] choose an SNC smooth projective compactification $\bar{X}$ of $X$. Work first on the open-and-closed part $X_d$ where the smooth morphism has relative dimension $d$; there are finitely many such parts, and their derived images sum to $R{f}_{\ast }\mathbb{Q}$. Each restriction remains smooth projective. The relatively ample Chern class is a morphism ${\mathbb{Q}}_{X_d}\to {\mathbb{Q}}_{X_d}[2]$: an injective resolution identifies degree-two cohomology with morphisms from the constant sheaf to its shift. Push it forward in [F6] and shift by $d$ to obtain $L:K\to K[2]$ for $K=R{f}_{\ast }{\mathbb{Q}}_{X_d}[d]$, whose stalk action is the cup-product Lefschetz action of [F1]. Now $H^{-a}(K)=R^{d-a}{f}_{\ast }{\mathbb{Q}}_{X_d}$ and $H^a(K)=R^{d+a}{f}_{\ast }{\mathbb{Q}}_{X_d}$, and $L^a$ between these sheaves is an isomorphism by fibrewise rational hard Lefschetz and proper base change [F2, F6]. Thus the centered hypothesis of [F2] applies to $K$, and shifting back and summing over $d$ gives $R{f}_{\ast }\mathbb{Q}\cong {⨁}_j{R}^j{f}_{\ast }\mathbb{Q}[-j]$. Precompose this isomorphism with the inverses of its induced automorphisms on the summands $R^j$ so that it induces the identity on every cohomology sheaf. Under the Leray identification $H^m(X;\mathbb{Q})\cong H^m(S,R{f}_{\ast }\mathbb{Q})$ supplied by [F6], this decomposition expresses $H^m(X;\mathbb{Q})$ as the direct sum of the groups $H^{m-j}(S,R^j{f}_{\ast }\mathbb{Q})$; the $j=m$ summand is a right inverse to the canonical edge map $H^m(X;\mathbb{Q})\to H^0(S,R^m{f}_{\ast }\mathbb{Q})$, since the normalized summand inclusion induces the identity on $R^m$; hence the edge map is surjective. Equivalently the normalized decomposition makes that edge map the projection onto this summand. The stalk of $R^m{f}_{\ast }\mathbb{Q}$ at $s$ is $H^m(X_s;\mathbb{Q})$ by proper base change [F6], and evaluation at $s$ from the global flat sections $H^0(S,R^m{f}_{\ast }\mathbb{Q})$ is injective because $S$ is connected; its image is exactly the monodromy-invariant subspace $H^m(X_s;\mathbb{Q}{)}^{{\pi }_1(S,s)}$. Consequently the composite $H^m(X;\mathbb{Q})\to H^m(X_s;\mathbb{Q})$ has image the invariant subspace.

Weights and strictness identify the invariant subspace. The space $H^m(X,\mathbb{Q})$ carries the mixed Hodge structure of [F3], whose weights are at least $m$; the target $H^m(X_s,\mathbb{Q})$ is pure of weight $m$, because $X_s$ is a smooth projective variety and the restriction $H^m(X,\mathbb{Q})\to H^m(X_s,\mathbb{Q})$ is a morphism of mixed Hodge structures by the smooth-open functoriality of [F3], applied to the algebraic fibre inclusion; the compact fibre structure agrees with its pure structure because its boundary is empty. Strictness [F4] applied to this morphism, whose target has $W_{m-1}=0$ and $W_m=H$, gives that the image of $H^m(X,\mathbb{Q})$ equals the image of its minimal weight part $W_m{H}^m(X,\mathbb{Q})$; by [F3] that minimal weight part is the image of $H^m(\bar{X},\mathbb{Q})\to H^m(X,\mathbb{Q})$. Combining with step 1.1, the image of $H^m(\bar{X},\mathbb{Q})\to H^m(X_s,\mathbb{Q})$ equals the image of $H^m(X,\mathbb{Q})\to H^m(X_s,\mathbb{Q})$, which is the monodromy-invariant subspace. Since $H^m(\bar{X},\mathbb{Q})$ and $H^m(X_s,\mathbb{Q})$ are pure of weight $m$ and the restriction is a Hodge morphism by [F1], its image is a Hodge substructure; this proves that the invariant subspace is a Hodge substructure in the family case.

Base-point independence and flatness of Hodge components in the family case. The restriction $H^m(\bar{X},\mathbb{Q})\to H^m(X_s,\mathbb{Q})$ factors through the canonical map $E:H^m(\bar{X},\mathbb{Q})\to H^0(S,R^m{f}_{\ast }\mathbb{Q})$ followed by evaluation at $s$, and $E$ is independent of $s$. Its kernel $K=\ker E$ is therefore one fixed rational subspace, and evaluation identifies $H^0(S,R^m{f}_{\ast }\mathbb{Q})$ with the quotient $H^m(\bar{X},\mathbb{Q})\mathrm{/}K$, carrying the quotient Hodge structure, at every base point simultaneously; this is the asserted one constant Hodge structure on the invariant spaces. Now let $v$ be a global flat section of $H_{\mathbb{C}}$. Since the invariant space at $s$ is a Hodge substructure by step 2.1, the Hodge components of the value $v_s$ lie again in the complexified invariant space, hence are values of global flat sections $w_1,\dots ,w_r$; their sum ${\sum }_i{w}_i$ is a global flat section with the same value at $s$ as $v$, so it equals $v$ by connectedness. Since the Hodge decomposition of the invariant fibre is the fixed quotient structure of step 2.1, the section $w_i$ is the Hodge component of $v$ of the corresponding type at every fibre. This proves, in the family case, both that the invariant subspace is a Hodge substructure and that the Hodge components of global flat sections are global flat sections.

Reduction of a general geometric variation to the realization open. Let $U\subseteq S$ be a nonempty Zariski open on which $H$ has an algebraic realization: $H{\mathrm{\mid }}_U$ is the image of a flat rational idempotent $p$ that is a Hodge morphism at every point, inside $A={⨁}_i{R}^{m_i}(f_i{)}_{\ast }\mathbb{Q}(r_i)$ for smooth projective $f_i:X_i\to U$; this is the definition of a geometric variation, and each summand is a finite direct sum of weight pieces. By steps 1.1-3.1 applied to each $f_i$ over $U$ (note $U$ is again connected smooth quasiprojective), the statement holds for every $R^{m_i}(f_i{)}_{\ast }\mathbb{Q}(r_i)$: its invariant part is a constant Hodge substructure and the Hodge components of its global flat sections are flat. Finite direct sums and Tate twists preserve both assertions, because monodromy acts componentwise and the twist changes only the weight; hence the statement holds for $A{\mathrm{\mid }}_U$. Because $p$ is flat and a Hodge morphism, it commutes with monodromy and with the Hodge decompositions, so the invariant space of $H{\mathrm{\mid }}_U$ is the image under $p$ of the invariant space of $A{\mathrm{\mid }}_U$, and it inherits the constant Hodge structure; the Hodge components of a flat section of $H{\mathrm{\mid }}_U$ are the images under $p$ of the components of a lift and are therefore flat. This proves the assertions for $H{\mathrm{\mid }}_U$.

Extension from $U$ to $S$. The inclusion $U\hookrightarrow S$ induces a surjection ${\pi }_1(U,s)\to {\pi }_1(S,s)$ by [F5]. Therefore a vector of $H_s$ fixed by ${\pi }_1(U,s)$ is fixed by ${\pi }_1(S,s)$, so restriction of flat sections $H^0(S,H)\to H^0(U,H{\mathrm{\mid }}_U)$ is an isomorphism: it is injective because $S$ is connected, and surjective by the preceding sentence. Apply this restriction isomorphism also after complexification. Given a global flat section $v$ of $H_{\mathbb{C}}$, its restriction to $U$ has, by step 4.1, Hodge components that are flat sections of $H_{\mathbb{C}}{\mathrm{\mid }}_U$; each extends uniquely to a global flat section $w_i$ of $H_{\mathbb{C}}$, and $v-{\sum }_i{w}_i$ is a global complex flat section vanishing on the dense open $U$, hence vanishes by continuity. For each type $(a,b)$, the corresponding $w_i$ takes its values in the continuous Hodge subbundle of that type on $U$. Membership in this subbundle is a closed condition: in a local trivialization its continuously varying projection has kernel describing the complement, and the difference between $w_i$ and its type projection is continuous. This difference vanishes on the dense open $U$, hence on all of $S$. Thus $w_i$ has the prescribed type everywhere; uniqueness of the fiber decomposition makes these global flat sections exactly the Hodge components of $v$. In particular the complexification of $H_s^{{\pi }_1(S,s)}$, consisting of values of global complex flat sections, is stable under all Hodge type projections. The rational invariant space is therefore a Hodge substructure with the induced pure decomposition.

Independence of the base point for a general variation. Evaluation identifies the rational space $E=H^0(S,H)$ with each invariant fiber $H_s^{{\pi }_1(S,s)}$. Transport along a path identifies these invariant fibers through this one space $E$, since a global flat section is transported to its own value. Step 5.1 proves that the Hodge components of every global flat section are again global flat sections. On the realization open $U$, steps 3.1 and 4.1 identify these type decompositions with one fixed quotient Hodge structure on $E$, and the closed-membership argument of step 5.1 extends those same types to every point of $S$. Consequently evaluation gives one Hodge structure on $E$ independent of $s$, and transport restricted to the invariant fibers preserves that structure. This assertion concerns the invariant constant subsystem; it does not assume that transport on the full variation preserves the moving Hodge decomposition. Finite sums of pure weights are handled one weight at a time, proving every clause of the statement.