Semisimplicity of geometric Gauss-Manin local systems

Rational descent of the socle and its lattice. Let $H$ be a nonzero pure geometric object of rank $r$ over the connected base $S$, and let $W_{\mathbb{C}}$ be the socle of the complex monodromy module $V_{s,\mathbb{C}}$, a nonzero finite-dimensional complex module by [F2]. By [F2] there is a rational flat subspace $W_{\mathbb{Q}}\subseteq V_s$ with $W_{\mathbb{C}}=W_{\mathbb{Q}}\otimes \mathbb{C}$; since it is monodromy-invariant it is the fibre at $s$ of a local subsystem of $H$, which we again call $W$. Let $H_{\mathbb{Z}}$ be the full monodromy-stable integral lattice of [F3] and put $W_{\mathbb{Z}}=W_{\mathbb{Q}}\cap H_{\mathbb{Z}}$; a rational basis of $W_{\mathbb{Q}}$ becomes integral after clearing denominators, so $W_{\mathbb{Z}}$ is a full lattice in $W_{\mathbb{Q}}$, and the monodromy preserves it and acts by invertible integral matrices. On the determinant line $\det W={\bigwedge }^{e}W$, $e={\dim }_{\mathbb{Q}}{W}_{\mathbb{Q}}$, the monodromy therefore acts by $\pm 1$, so $(\det W{)}^{\otimes 2}$ is a constant rank-one local subsystem of the geometric tensor object ${\bigwedge }^{e}H\otimes {\bigwedge }^{e}H$ of [F3], generated by a global horizontal section.

Torus images of simple summands are flat. Write the complex socle as $W_{\mathbb{C}}={⨁}_i{S}_i$ by [F4]. Fix a summand $T$ of dimension $d$ and its flat complement $T^{\mathrm{\prime }}={⨁}_{S_i\mathrm{\ne }T}{S}_i$ in $W_{\mathbb{C}}$, using the chosen summand indices. Put $A={\bigwedge }^d{H}_{\mathbb{C}}\otimes {\bigwedge }^{e-d}{H}_{\mathbb{C}}$ and $V={\bigwedge }^{d}T\otimes {\bigwedge }^{e-d}{T}^{\mathrm{\prime }}\subset A$. The determinant identification for $W=T\oplus T^{\mathrm{\prime }}$ is flat, so step 1.1 makes $V^{\otimes 2}\subset A^{\otimes 2}$ a constant line. The rational object underlying $A^{\otimes 2}$ is geometric by [F3]; its complex invariant part is the complexification of its rational invariant part, because invariants are common kernels of rational matrices, reducible to finitely many linear equations in a finite-dimensional fibre. By [F1] this fixed part is a constant Hodge structure. For any Hodge-torus element $t$, the image $t(V^{\otimes 2})=(tV{)}^{\otimes 2}$ is therefore a flat line in that fixed part. In a local flat frame choose a nonzero continuous generator $\sigma$ of $tV$; its squared projective class is constant. The map $[u]\mapsto [u\otimes u]$ is injective: if $u\otimes u$ and $v\otimes v$ are proportional, contraction with a functional nonzero on $u$ shows $v$ is proportional to $u$. Thus $[\sigma ]$ is constant and $tV$ is flat. A decomposable tensor line in $E\otimes F$ determines its factor lines by contraction with $F^{\ast }$ and $E^{\ast }$; applying this to $tV$ shows that ${\bigwedge }^d(tT)$ is flat. Finally the Plücker line determines $tT$ by $tT=\{v:v\wedge p=0\}$ for any nonzero $p\in {\bigwedge }^d(tT)$, as is checked by completing a basis of $tT$ to a basis of the ambient space. This identity commutes with transport, so $tT$ is flat for every $t$. The case $e=d$ uses the constant zeroth exterior power and satisfies the same argument.

Openness of irreducibility and torus-stability of the socle. For $t$ near $1$ the flat subspace $tT$ is simple: were there $t_n\to 1$ with $t_{n}T$ reducible, there would be, for each $n$, a proper nonzero flat subspace $U_n\subset t_{n}T$; their dimensions have finitely many possible values $\varrho$, $1\le \varrho <d$; pass to a subsequence with fixed $\varrho$, then use [F5] to make the orthogonal projections $P_n$ onto $U_n$ converge to a rank-$\varrho$ projection $P$. The inclusions $U_n\subset t_{n}T$ give $\mathrm{im}P\subset T$. For each monodromy matrix $M$, invariance is the closed matrix equation $(I-P_n)M{P}_n=0$; taking limits gives $(I-P)MP=0$ for every $M$. No finite generation of the fundamental group is assumed. Thus $U=\mathrm{im}P$ is a nonzero proper flat subspace of $T$, contradicting simplicity of $T$. Hence, for $t$ in a neighbourhood of $1$, each $t{S}_i$ is a simple flat subspace of $H_{\mathbb{C}}$, and every simple flat subspace is contained in the socle $W_{\mathbb{C}}$ by [F2]; summing over the finitely many $S_i$ gives $tW\subseteq W$, hence $tW=W$ by dimension. Thus the stabilizer of $W$ in ${\mathbb{C}}^{\times }$ contains a neighbourhood of the identity; a subgroup of a topological group with nonempty interior is open, and the connected group ${\mathbb{C}}^{\times }$ has no proper nonempty open subgroup, so $tW=W$ for every $t\in {\mathbb{C}}^{\times }$. Consequently $W_s$ is stable under the Hodge torus at every point $s$, so $W_{\mathbb{Q}}$ is a Hodge substructure of $H$ at every point; since it is already flat, it is a Hodge subvariation.

Polarization splitting and induction. Let $Q$ be a polarization of the pure object $H$. Its restriction to the Hodge subvariation $W$ is nondegenerate: if $v\in W_{s,\mathbb{C}}$ were a nonzero radical vector, then with the Weil operator $C$ of the Hodge structure the vector $C\bar{v}$ lies in $W_{s,\mathbb{C}}$ because $W_s$ is a Hodge substructure, so the positivity of the Weil-operator Hermitian form $h(v,v)=Q(Cv,\bar{v})=(-1{)}^{n}Q(v,C\bar{v})$ would give the contradiction $0=Q(v,C\bar{v})\mathrm{\ne }0$. Hence the $Q$-orthogonal complement $W^{\perp }$ is a flat Hodge subvariation and $H=W\oplus W^{\perp }$; the orthogonal projection onto $W$ is a flat rational idempotent that is a Hodge morphism, so $W^{\perp }$ is a geometric direct summand of $H$ by [F3], of smaller rank unless $W=H$. The submodule $W_{\mathbb{C}}$ is a direct sum of simples by construction, and by induction over the rank the complex monodromy module of $W^{\perp }$ is semisimple; a direct sum of semisimple modules is semisimple, so the complex monodromy module of $H$ is semisimple. A non-pure object is a finite direct sum of pure weight pieces, handled one weight at a time, and the rank-zero case is the empty direct sum.

Coefficient fields and the Gauss-Manin case. Complex semisimplicity of the monodromy module of $H$ means exactly that the complex local system $H\otimes \mathbb{C}$ is semisimple in the sense of [F4]. By [F6] the rational local system $H$ is then semisimple over $\mathbb{Q}$ and over every characteristic-zero field, with no embedding of that field into $\mathbb{C}$ used. For a smooth projective $f:X\to S$, the local systems $R^m{f}_{\ast }\mathbb{Q}$ are geometric and polarizable by [F3], so step 4.1 and [F6] apply to give their semisimplicity over $\mathbb{Q}$ and over every characteristic-zero field. This proves both asserted clauses; the algebraic realization and the integral lattice were used exactly in steps 1.1 and 2.1, and no statement is made about an arbitrary polarized analytic variation on a punctured disk.