Lemma 11 ([Period quotient and descent]). There is an algebraic quotient with connected generic fibers which remembers the full integral period of g and the compact marks. Both g and 𝔸 descend as variations on a dense smooth open of the …

Use the full filtration on the rational integral polarized Lie variation, not only the highest subspace of 𝔸 . Let Γ be an arithmetic group of isometries of its polarized lattice containing its monodromy, and let Ω be its period domain, or the associated generic Mumford--Tate domain. The algebraicity theorem for pure integral period maps factors the period map through a dominant algebraic map to a quasi-projective algebraic image. Its hypotheses hold here by Lemma 7; in particular a lattice has been retained in the full Lie system. Normalize this image in the relative algebraic closure of its function field in (B) . The resulting map b_0:B^ T^ has geometrically connected generic fiber. Shrink to smooth opens where it is a morphism. The dimension of T^ is the generic rank of the full period map. Indeed, the real period stabilizer is compact, arithmetic orbits are discrete, and the factorization preserves the local analytic image dimension. Here is the local-system descent, including possible finite monodromy. Compactify the graph properly over T^ , resolve the complement, and shrink T^ so that the compactification and all strata of its SNC complement are smooth over it. Lifting vector fields tangent to the strata gives a local differentiable trivialization of this proper smooth pair and hence of its open part. We may thus assume that b_0 is a topological fiber bundle with connected fibers. Removing a proper algebraic subset of a smooth complex variety induces a surjection on its fundamental group, so these shrinkings do not decrease the relevant monodromy images. Choose a Hodge-generic point of B^ . Along the universal cover of its b_0 -fiber the lifted Lie-period map has values in one discrete arithmetic orbit; it is therefore constant. Every vertical loop acts by a rational Lie automorphism fixing that lifted period.Lemma 10 ([Generic Lie stabilizer]) makes its action trivial. The exact sequence π_1(b_0^ -1 (t))⟶ π_1(B^)⟶π_1(T^) ⟶1 now descends the Lie representation. The representation on 𝔸 descends as well: by Lemma 8 ([Homogeneous replacement]) it factors through the full projected Lie action, including its component group. This is where removal of the finite scalar image is necessary. The Lie filtration is constant in flat coordinates along each connected fiber. The derivative of the filtration on 𝔸 is the action of ; it too vanishes on those fibers. Consequently both filtrations descend. Holomorphicity, Griffiths transversality, and polarization can be checked after the smooth surjective pullback. The descended local systems have quasi-unipotent local monodromy: for a boundary valuation of T choose a valuation above it on a proper model of B ; a positive power of its monodromy is a monodromy of the original variation. Quasi-unipotence of that power implies quasi-unipotence of the original operator. Canonical extension and GAGA, or regular-singular Riemann--Hilbert, therefore give the algebraic flat bundles and algebraic Hodge bundles on these opens; see and . Over T^ consider the algebraic flag bundle of the descended compact factors. The section j_ u gives an algebraic map from B^ into that bundle. Take its image and again normalize in the relative algebraic closure inside (B) . This is the desired S^ . Variations pull back from T^ , and the compact flag descends by its tautological definition. On a dense open the finite map from S^ to its image is étale, and the period-image factorization preserves tangent rank. Hence the full period and compact marks have joint differential of rank S . Resolve projective compactifications and the map from B to obtain b:B→S with connected fibers. The monodromy images on the open source and target agree by the fundamental-group sequence just used. For a monodromy-stable collection of rational factors the projected Lie system again has its rational polarization and lattice. The identical argument gives its quotient and descent. At connected level every rational normal factor collection is permitted.