We study the interaction between two structures on the relative de Rham moduli of flat bundles in a smooth proper family: on the one hand, the condition, which fixes the topological (Betti) monodromy and defines a horizontal foliation; on the other hand, constraints, which are expected to force such horizontal data to be algebraic. Our guiding principle is that, much as the Grothendieck–Katz philosophy ties integrality of a linear differential equation (via p-curvature) to finiteness of its monodromy, an appropriate integrality condition on a isomonodromy deformation should force algebraicity of the corresponding leaf.
Concretely, the isomonodromy foliation on MdR(X/S, r) is defined by the condition that the associated Betti representation (after choosing an embedding into C and applying Riemann–Hilbert) remains constant up to conjugacy as one moves in the base. Formally, through a point corresponding to a flat bundle on a fiber, there is a canonical horizontal deformation over the completion ŜK, s, obtained by restricting the universal isomonodromy crystal. A priori this deformation is defined only over the fraction field K and only formally; the fundamental input in this work is an refinement: after inverting a single element N ∈ R, the formal deformation descends to the arithmetic completion ŜR, s[1/N]. We refer to this as . It should be understood as a nonlinear avatar of the existence of a good model at almost all primes, and it is this input that allows us to import p-adic and mod-p arguments into the study of isomonodromy leaves.
The basic class of flat bundles for which such an integrality statement is known is provided by Picard–Fuchs (Gauss–Manin) connections. If (ℋ, ∇GM) arises as (Riπ*ΩY/U•, ∇GM) for a smooth proper morphism π : Y → U, then ℋ is constructed functorially from algebraic de Rham cohomology and therefore carries additional structure: Hodge filtrations in characteristic 0, and, after reduction mod p away from finitely many primes, conjugate filtrations and Cartier descent data. Lam–Litt establish an integrality theorem for the isomonodromy deformation attached to such Picard–Fuchs objects, which may be viewed as a nonlinear counterpart of Katz’s analysis of p-curvature for Gauss–Manin systems. In the present work, we use this Picard–Fuchs integrality as an input and ask how far it propagates under natural operations, in particular under passage to semisimple direct summands.
The necessity to treat direct summands is not merely technical. From the viewpoint of geometry and arithmetic, the most interesting local systems are rarely the full Gauss–Manin system; rather, they are constituents singled out by symmetry or by motivic considerations: eigenspaces under a deck group for cyclic covers, factors cut out by endomorphisms of an abelian scheme, or more generally pieces predicted by the theory of motivated cycles. On the Betti side, these appear as semisimple summands of a geometric local system; on the de Rham side, they appear as semisimple objects in MIC(U/R) that are by idempotent projectors. The difficulty is that, even if a summand exists abstractly in characteristic 0, it need not be defined integrally: idempotents may involve denominators, and, without control of these denominators, the summand may fail to extend well at primes of bad reduction. Since the Lam–Litt method is fundamentally integral (it compares characteristic 0 and characteristic p filtrations), it is not automatic that its conclusions descend to arbitrary semisimple factors.
Our main results show that, under a motivic hypothesis ensuring that
the summand projector is induced by a Chow correspondence, the Lam–Litt
integrality mechanism . More precisely, we assume that on a dense open
U ⊆ Xs
our given flat bundle (E, ∇)|U is the
image of a acting on a Gauss–Manin connection, and that this idempotent
comes from an algebraic correspondence
p ∈ CHdim (Y/U)(Y×UY) ⊗ Q,
so that it has good functoriality properties with respect to de Rham,
Hodge, and crystalline realizations. Under this hypothesis, integrality
of the formal isomonodromy deformation forces the corresponding
isomonodromy leaf to be algebraic; equivalently, the Betti
representation has finite orbit under the outer monodromy action of
π1(S(C)an, s)
on the appropriate character variety.
We now outline the mechanism behind this implication.
The bridge between integrality and finiteness is built using filtrations. For Gauss–Manin connections, one has the Hodge filtration F• on the de Rham bundle in characteristic 0, and, after reduction mod p, the conjugate filtration C• on the corresponding object in characteristic p. Lam–Litt prove that, for a Picard–Fuchs system, integral formal isomonodromy implies that these filtrations extend along the formal isomonodromy leaf in a way compatible with the connection (Griffiths transversality in characteristic 0, and the appropriate Frobenius–Cartier compatibility in characteristic p). In informal terms, integrality of the nonlinear deformation forces the local system to carry an ``integral Hodge-theoretic structure’’ along the entire formal leaf.
In order to use this for a summand (E, ∇)|U ⊆ ℋ, we need to show that the filtration extension on ℋ restricts to E. This is where the motivated correspondence hypothesis enters: correspondences act functorially on cohomology and preserve the Hodge filtration; moreover, away from finitely many primes, they also act on crystalline realizations and preserve conjugate filtrations. Thus the idempotent projector cutting out E can be arranged (after inverting one element of R) to exist integrally and to act on the Gauss–Manin system, both in characteristic 0 and mod p.
At the formal level, this leads to a simple but crucial observation: if $(\widehat{V},\widehat{\nabla})$ is a formal flat bundle equipped with a Griffiths-transverse filtration F• and if e is a horizontal idempotent preserving F•, then F• restricts to a Griffiths-transverse filtration on Im(e). In other words, once the projector is known to be compatible with the filtration, the filtration extension automatically passes to the direct summand. This formal ``summand-stability’’ is the point at which we exploit semisimplicity: we can treat our object as a direct factor and avoid extension data.
The second step is a finiteness principle of Deligne–Simpson type. Very roughly, if a complex local system underlies a polarizable Z-variation of Hodge structure with fixed numerical invariants, then there are only finitely many possibilities up to isomorphism on a fixed base; equivalently, the corresponding point in the Betti moduli has finite orbit under the action of π1(S) induced by monodromy in the family. For our purposes, the relevant input is that the filtration extension along the isomonodromy deformation produces precisely the additional structure needed to place each π1(S)-conjugate of the Betti representation into an integral Hodge-theoretic category, to which Deligne’s finiteness applies. This converts the Hodge-theoretic extension statement into an honest finiteness statement on the Betti side.
At this stage, one passes from finiteness of orbit to algebraicity of the isomonodromy leaf using the Riemann–Hilbert correspondence and the standard identification of the isomonodromy foliation with the locus of locally constant Betti monodromy. In particular, finite π1(S)-orbit is equivalent to the statement that the horizontal section of the de Rham moduli determined by (E, ∇) factors through a finite (hence algebraic) cover of a Zariski-open neighborhood of s ∈ S.
It is worth emphasizing that, although semisimplicity guarantees the existence of abstract idempotents in characteristic 0, it does not control their arithmetic behavior. If the projector defining a summand requires denominators varying with p, then even if the ambient Gauss–Manin system has good integral properties, the summand may fail to carry any uniform integral structure at almost all primes. In that situation one cannot expect mod-p filtrations to restrict correctly, and the Lam–Litt arguments do not apply. The motivated correspondence hypothesis supplies precisely the missing uniformity: Chow correspondences spread out over finitely generated Z-algebras, have bounded denominators after inverting a single N, and act compatibly on all cohomological realizations. In concrete examples, such projectors come from group algebra idempotents (cyclic covers), from endomorphism algebras of abelian schemes, or from motivated cycles in the sense of André.
Under the hypotheses fixed in the enclosing scope, we prove that integral formal isomonodromy for a semisimple motivated Picard–Fuchs summand forces algebraicity of the isomonodromy leaf through the corresponding point of MdR(X/S, r), and hence finiteness of the π1(S)-orbit of the associated Betti representation. This is the content of Theorem~A in the global context.
We also formulate a Tannakian upgrade. Let ⟨PF⟩ denote the smallest neutral Tannakian subcategory of MIC(U/K) generated by Gauss–Manin systems and closed under semisimple subquotients. The same strategy applies inductively through the Tannakian operations (tensor products and duals preserve filtration extension along an integral isomonodromy leaf, while semisimple subquotients are handled by the motivated summand argument), yielding a statement for any semisimple object in ⟨PF⟩ provided the relevant idempotents are realized by motivated correspondences after inverting a single N. This is Theorem~B.
Finally, in the case where X → S is a family of curves (or punctured curves in a logarithmic variant), the finiteness of the π1(S)-orbit admits a geometric re-interpretation as finiteness of mapping class group or braid group orbits in the corresponding character variety. Thus the same integrality criterion yields finiteness statements for surface group representations arising from motivated Picard–Fuchs summands, as recorded in Corollary~C.
The proof is organized to isolate the three logically distinct inputs.
First, we invoke the Lam–Litt integrality theorem for Picard–Fuchs systems to obtain extension of the relevant filtrations along the integral formal isomonodromy deformation for the ambient Gauss–Manin connection ℋ.
Second, we show that motivated idempotent projectors spread out after inverting a single element of R and act compatibly with Hodge and conjugate filtrations. This allows us to apply a general summand-stability lemma for filtrations along formal isomonodromy leaves, thereby producing a filtration extension for the summand (E, ∇)|U itself.
Third, we translate filtration extension into finiteness of the Betti orbit via Deligne–Simpson finiteness and Simpson’s non-abelian Hodge-theoretic framework, and then identify this finiteness with algebraicity of the isomonodromy leaf through the Riemann–Hilbert correspondence.
In this way the argument separates the Picard–Fuchs integrality input (which is specific and deep) from the formal summand-stability mechanism (which is elementary once stated correctly) and from the finiteness principle (which is Hodge-theoretic and topological). The resulting criterion applies uniformly to a wide range of geometric local systems that occur as motivated constituents of Gauss–Manin systems, including those appearing in explicit constructions such as cyclic covers and in arithmetic settings such as families of abelian varieties.
In this section we fix a smooth proper morphism f : X → S with geometrically connected fibers, and we recall the two moduli problems attached to rank r flat bundles on the fibers of f: the de Rham moduli stack MdR(X/S, r) and the Betti moduli stack of representations. We then explain how the condition appears as a horizontal (crystal) structure on MdR(X/S, r), and we record the basic equivalence between algebraicity of an isomonodromy leaf and finiteness of the induced π1(S)-orbit on Betti moduli after passage to C.
Let T → S be an
S-scheme, and write XT := X×ST.
When XT → T
is smooth, we denote by MIC(XT/T)
the groupoid of pairs (ℰ, ∇) where
ℰ is a vector bundle on XT of rank r and
∇ : ℰ → ℰ⊗𝒪XTΩXT/T1
is an integrable T-connection.
(We will not impose additional conditions such as fixed determinant,
although doing so produces variants with better separatedness
properties.) The formation of MIC(XT/T)
is compatible with base change in T in the usual way: given T′ → T over
S, pulling back (ℰ, ∇) along XT′ → XT
yields an object of MIC(XT′/T′).
We define MdR(X/S, r) to be
the stack over S whose fiber
over T → S is the
groupoid MIC(XT/T)
(with morphisms given by horizontal bundle isomorphisms). In particular,
for a point s ∈ S we
have a fiber stack
MdR(Xs, r) := MdR(X/S, r)×Ss,
which is the de Rham moduli stack of rank r flat bundles on the smooth proper
scheme Xs.
Although MdR(X/S, r) is in
general an Artin stack rather than a scheme, it carries the basic
geometric structures one expects: it is locally of finite presentation
over S, and its deformation
theory is governed by the de Rham complex with coefficients in the
endomorphism bundle. Concretely, if (ℰ, ∇) is a flat bundle on a fiber Xs, then
first-order deformations are controlled by
H1 (Xs, End(ℰ) ⊗ ΩXs•),
and obstructions lie in H2 of the same
complex. We will only use this deformation-theoretic picture as
motivation for the existence of a canonical horizontal formal
deformation discussed below; we will not need to impose smoothness of
the moduli stack at the point of interest, except when we speak
informally about ``leaves’’ of a foliation.
After choosing an embedding into C, we can describe the
corresponding Betti moduli problem in topological terms. Let Xs(C)an
be the complex analytic manifold associated to the fiber, and fix a
basepoint. The Betti moduli stack of rank r local systems on Xs(C)an
may be presented as the representation stack
MB(Xs, r) = [Hom(π1(Xs(C)an), GLr)/GLr],
where GLr acts by
conjugation. This stack (or its GIT coarse moduli space on the
semisimple locus) is the natural receptacle for monodromy
representations.
The family f : X → S induces an action of π1(S(C)an, s) on π1(Xs(C)an), well-defined up to inner automorphism, coming from the long exact sequence of homotopy for a topological fibration after restricting to a suitable Zariski-open subset of S over which f is a C∞-fiber bundle. Passing to representations modulo conjugation, we obtain an induced action of π1(S(C)an, s) on MB(Xs, r) (and on its coarse moduli on semisimple points). We refer to this as the .
The dynamical statement that will later be relevant is finiteness of the orbit of a given Betti point [ρ] ∈ MB(Xs, r)(C) under this action. Since we work with stacks, ``orbit’’ is understood up to isomorphism, i.e. we consider the set of isomorphism classes of representations obtained by precomposing ρ with the outer action of π1(S), and then passing to conjugacy classes.
For a fixed complex smooth proper variety Xs, the
Riemann–Hilbert correspondence identifies algebraic flat bundles with
regular singular analytic connections, and for smooth proper varieties
no singularity issues occur: any algebraic flat connection analytifies
to a holomorphic flat connection, whose horizontal sections form a
complex local system. Conversely, any complex local system arises from a
holomorphic flat bundle, and by GAGA the latter is algebraic. We
therefore obtain, on points,
(ℰ, ∇) ↦ ker(∇an),
which induces an equivalence between the groupoid of algebraic flat
bundles and the groupoid of complex local systems.
On moduli, this correspondence globalizes to an analytic isomorphism
between the de Rham and Betti moduli stacks. More precisely, after base
change to C and
analytification, there is an analytic equivalence of stacks
MdR(Xs, r)an ≃ MB(Xs, r)an,
compatible with pullback functorialities and with the restriction to
semisimple loci (where one may also pass to coarse moduli spaces if
desired). We will freely use this identification to translate statements
about isomonodromy in de Rham moduli into orbit statements in Betti
moduli.
For a family X → S the situation is more subtle: the Betti moduli of a single fiber does not itself vary algebraically with s, but rather it comes with the outer monodromy action described above. The de Rham moduli vary algebraically over S, and the isomonodromy condition is precisely the requirement that, under Riemann–Hilbert, the associated Betti class be constant in the appropriate sense. Making this precise leads to the crystal viewpoint.
We now recall the ``horizontal’’ structure on MdR(X/S, r). The key observation is that integrable connections satisfy a rigidity property with respect to nilpotent thickenings: they form a crystal in groupoids.
Let T be an S-scheme and let i : T ↪ T′
be a nilpotent thickening over S (i.e. a closed immersion defined
by a nilpotent ideal). Since X → S is smooth, the base
change XT′ → T′
is smooth, and one can ask how objects of MIC(XT/T)
compare with objects of MIC(XT′/T′)
restricting to them. The standard infinitesimal theory of integrable
connections shows that restriction induces an equivalence of
groupoids
$$
\mathrm{MIC}(X_{T'}/T')\ \xrightarrow{\ \sim\ }\ \mathrm{MIC}(X_T/T),
$$
functorial in (T ↪ T′).
Equivalently, the stack MdR(X/S, r) is a
on the infinitesimal site of S.
We interpret this crystal structure as the algebro-geometric avatar of isomonodromy. Informally, a ``horizontal’’ deformation over a nilpotent thickening is one that is uniquely forced by its restriction to the reduced base, and the crystal property says that deformation in MdR(X/S, r) is of this type. The nontrivial content is that, although MdR(X/S, r) may have many directions in which it varies over S, its variation is constrained by this canonical infinitesimal identification, which behaves like a flat connection on the moduli stack.
Specializing to the formal neighborhood of a point makes the
isomonodromy deformation concrete. Fix s ∈ S and consider the
formal completion Ŝs of S at s. Given a point
[(Xs, E, ∇)] ∈ MdR(X/S, r)(κ(s)),
the crystal property implies that for each Artinian local S-algebra A mapping to Ŝs, there is a
object (EA, ∇A) ∈ MIC(XA/A)
restricting to (E, ∇) on the
special fiber. These compatible objects assemble into a formal flat
bundle over the formal scheme Ŝs, which we
call the of (E, ∇) at s. By construction it is a
horizontal section of the crystal MdR(X/S, r) over
Ŝs.
If we work over the fraction field K of R and base change to SK, we similarly obtain a formal section over ŜK, s through the corresponding K-point of the moduli stack. This is the version that is relevant for arithmetic questions, since it is initially defined only over K and one may ask whether it descends to an integral model after inverting finitely many primes.
To pass from the formal picture to an algebro-geometric statement, we consider the foliation induced by the crystal. On a smooth scheme, a flat connection produces an integrable subbundle of the tangent bundle; on a moduli stack, one obtains an analogous integrable distribution on the smooth locus. We use the term for this distribution on MdR(X/S, r).
A (germ of a) through a point is, by definition, the maximal connected substack (in an analytic or formal neighborhood) obtained by integrating this distribution. Concretely, one may think of it as the image of the horizontal section of the crystal through the point. When one restricts to the formal completion at a point, the ``leaf’’ is literally the formal isomonodromic deformation constructed above.
The notion that will matter is when such a leaf is algebraic over the base.
This definition is tailored to the application: finiteness over S∘ is the precise condition ensuring that the leaf is cut out by algebraic equations in the ambient moduli and has no transcendental ``spread’’ in the base direction.
We now explain the equivalence between algebraicity of the isomonodromy leaf and finiteness of the Betti orbit after passage to C. Since the statement is standard in the form we need, we only sketch the mechanism and indicate where Riemann–Hilbert enters.
Assume temporarily that K ↪ C is fixed, so that we may view SC := S⊗RC and XC := X⊗RC as complex algebraic varieties, and we fix s ∈ S(C). Let (E, ∇) be a flat bundle on Xs and let ρ denote the associated Betti representation. The isomonodromy leaf through (E, ∇) consists, analytically, of those de Rham points whose Betti images lie in the π1(S(C)an, s)-orbit of [ρ] ∈ MB(Xs, r) under outer monodromy. Indeed, transporting (E, ∇) along a loop in S(C)an changes the identification of the fiber group π1(Xs) by an outer automorphism; on the de Rham side, the crystal provides the corresponding horizontal transport in moduli, and Riemann–Hilbert identifies the two transports.
We record the resulting equivalence in the form that will be used later.
We emphasize that Lemma~ is a comparison statement: the de Rham foliation becomes, under Riemann–Hilbert, the dynamical system given by the outer monodromy action on the Betti moduli of a fiber. Thus any algebro-geometric criterion producing algebraicity of a leaf immediately yields a finiteness statement for the corresponding representation, and conversely any independent finiteness statement on the Betti side can be interpreted as algebraicity of the associated isomonodromy locus on the de Rham side.
Finally, we note that although we formulated the discussion over C to invoke Riemann–Hilbert directly, the crystal construction of the formal isomonodromy deformation is algebraic and makes sense over general bases. The arithmetic problem we will pursue later is to recognize when the formal horizontal section defined over K actually descends to an integral formal model over ŜR, s[1/N], and then to convert this integrality into algebraicity of the leaf via the finiteness criterion above.
We now isolate the class of flat bundles to which our arithmetic input will apply. The guiding idea is that the connections that arise from geometry carry additional structure (Hodge filtrations in characteristic 0, conjugate filtrations in characteristic p) that can be propagated along isomonodromic deformations once one has an integral model. In order to descend such structure from a geometric object to a given flat bundle (E, ∇), we impose that (E, ∇) appear as a direct summand of a Gauss–Manin system cut out by an algebraic correspondence.
Let U be a smooth R-scheme and let π : Y → U be
smooth and proper. For each i ≥ 0 we consider the relative de
Rham complex ΩY/U•
and its higher direct images
ℋdRi(Y/U) := Riπ*ΩY/U•.
By standard results on smooth proper morphisms, ℋdRi(Y/U)
is locally free of finite rank on U, compatible with arbitrary base
change on U, and its formation
commutes with restriction to open subschemes. The defining feature is
that ℋdRi(Y/U)
carries a canonical integrable connection, the Gauss–Manin
connection
∇GM : ℋdRi(Y/U) → ℋdRi(Y/U)⊗𝒪UΩU/R1,
which we view as an object (ℋ, ∇GM) ∈ MIC(U/R).
We recall briefly one construction, mainly to emphasize
functoriality. The exact sequence of K"ahler differentials
0 → π*ΩU/R1 → ΩY/R1 → ΩY/U1 → 0
induces a filtration on ΩY/R•
whose associated graded is ΩY/U• ⊗ π*ΩU/R•.
Passing to the corresponding spectral sequence yields a canonical edge
morphism which, after taking Riπ*,
produces ∇GM; integrability
is a formal consequence of d2 = 0 and functoriality
of the construction. We will refer to such objects (ℋ, ∇GM) as .
In the situation of the enclosing scope, we allow ourselves to shrink from Xs to a dense open U ⊆ Xs in order to realize the restriction (E, ∇)|U inside such a Gauss–Manin system. The reason for working on a dense open is that the geometric realization of a given local system or flat bundle often becomes available only after removing a boundary divisor; we will not need a detailed analysis of singularities here, since our arguments concern the behavior along the base S rather than along Xs \ U.
Let d = dim (Y/U), so
that Y×UY is
smooth over U of relative
dimension 2d. A cycle
class
p ∈ CHd(Y×UY) ⊗ Q
may be viewed as a correspondence from Y to itself over U. On relative cohomology,
correspondences act by the usual pull–push formalism. On de Rham
cohomology this action is algebraic and compatible with the Gauss–Manin
connection.
Concretely, write pr1, pr2 : Y×UY → Y
for the projections. The de Rham cycle class of p defines a cohomology class
cldR(p) ∈ HdR2d(Y×UY/U)
and hence, for each i, an
endomorphism of ℋdRi(Y/U)
given fiberwise by
pdR(x) = (pr2)* (cldR(p) ∪ pr1*(x)), x ∈ ℋdRi(Y/U),
where (pr2)*
denotes the Gysin pushforward in relative de Rham cohomology. This
construction globalizes to an 𝒪U-linear endomorphism
pdR ∈ End𝒪U(ℋdRi(Y/U)).
The key property for us is horizontality:
∇GM ∘ pdR = (pdR ⊗ 1) ∘ ∇GM.
There are several equivalent ways to justify this. One may argue by
functoriality of ∇GM with
respect to maps of smooth proper U-schemes and then extend to
correspondences by linearity and the pull–push description above.
Alternatively, one may view (ℋdRi(Y/U), ∇GM)
as the realization of the relative cohomology object in the category of
crystals over U, in which case
correspondences define endomorphisms of the underlying crystal and
horizontality is built in. In either interpretation, the conclusion is
that pdR is an
endomorphism in MIC(U/R).
When p is idempotent, i.e. p ∘ p = p in CHd(Y×UY) ⊗ Q, the induced endomorphism pdR is an idempotent horizontal endomorphism of the Picard–Fuchs connection. This is the mechanism by which we cut out direct summands in MIC(U/R).
Suppose now that p is an
idempotent correspondence and let (ℋ, ∇GM) denote (ℋdRi(Y/U), ∇GM).
Since pdR is
horizontal and idempotent, it defines a decomposition in MIC(U/R) after passing to
images and kernels. More precisely, set
ℋp := Im(pdR) ⊆ ℋ, ℋ1 − p := Im(1 − pdR) ⊆ ℋ.
Because pdR is
𝒪U-linear and
idempotent, ℋp and
ℋ1 − p are locally
direct summands of ℋ; in particular
they are locally free, and we have ℋ ≃ ℋp ⊕ ℋ1 − p
as vector bundles on U.
Horizontality implies that ∇GM preserves each summand, so
ℋp inherits an
integrable connection. Thus ℋp is naturally an object
of MIC(U/R), and the
inclusion ℋp ↪ ℋ
and projection ℋ ↠ ℋp are horizontal
morphisms.
We say that a flat bundle (E, ∇)|U is a of the Picard–Fuchs connection ℋ if there exists such an idempotent correspondence p with (E, ∇)|U ≃ (ℋp, ∇GM|ℋp). The point is that p is not merely an abstract idempotent in EndMIC(ℋ), but comes from an algebraic cycle on Y×UY. This additional input will later control denominators and make it possible to compare characteristic 0 and characteristic p filtrations.
We stress that the existence of a de Rham idempotent does not, by itself, guarantee good arithmetic behavior. Even if (E, ∇)|U is known to be a direct summand of ℋ in MIC(U/K), the idempotent cutting out the summand can a priori have denominators depending on the point of U, and it need not spread to an endomorphism over R away from finitely many primes. Our hypothesis packages precisely the extra ``motivic’’ rigidity needed to rule out such pathologies: the projector is realized by a correspondence defined over R (after possibly inverting one element of R), hence admits compatible realizations in the cohomology theories that will intervene later.
The Picard–Fuchs connection carries the Hodge filtration in
characteristic 0. We recall the minimal
facts that will be used subsequently. The stupid filtration on ΩY/U•
induces a filtration by subbundles
Fpℋ ⊆ ℋ, Fpℋ = Im(Riπ*ΩY/U ≥ p → Riπ*ΩY/U•),
and ∇GM satisfies Griffiths
transversality:
∇GM(Fpℋ) ⊆ Fp − 1ℋ ⊗ ΩU/R1.
For our purposes, the crucial point is functoriality: the formation of
F• is compatible
with morphisms induced by algebraic correspondences. In particular, if
p is a correspondence as
above, then pdR
preserves F•ℋ.
Hence F• restricts
to a filtration on ℋp = Im(pdR),
and the associated graded pieces grFp(ℋp)
are direct summands of grFp(ℋ).
We formulate this functoriality now because it is the precise mechanism by which ``filtration extension’’ statements, proved first for the ambient Gauss–Manin system, will be transferred to the motivated summand. The transfer does not require any transcendental argument: it is a formal consequence of the fact that pdR is horizontal and filtered.
While we will not yet discuss characteristic p realizations, we note that the same functoriality holds for the conjugate filtration on de Rham (or crystalline) cohomology after reduction modulo almost all primes, again because correspondences act functorially. This is one of the reasons for insisting that our projector be motivic: it supplies a single algebraic object whose realizations simultaneously control the relevant filtrations in different characteristics.
The second aspect of our hypothesis concerns semisimplicity. Over C, Gauss–Manin connections coming from smooth proper morphisms underlie polarizable variations of Hodge structure; by Deligne’s semisimplicity theorem, the associated complex local systems are semisimple. Consequently, after base change to C, the Betti representation attached to ℋ is semisimple, and any direct summand cut out by a projector in the category of variations of Hodge structure is likewise semisimple.
However, in the present algebro-arithmetic setting, several distinctions matter. First, (E, ∇) is given a priori only on the fiber Xs, and we realize it as a summand only after restricting to a dense open U ⊆ Xs. Second, even when a summand exists in MIC(U/K), semisimplicity is a property of the associated Betti local system after choosing an embedding K ↪ C, and it does not automatically follow from the existence of an abstract projector in MIC(U/K) unless one knows compatibility with the polarization/Hodge structure on the ambient Gauss–Manin system. The motivic nature of the correspondence addresses this compatibility, but we will not attempt to reprove semisimplicity from it.
We therefore impose semisimplicity as part of the input: we assume (E, ∇)|U is semisimple in MIC(U/R) (equivalently, after embedding into C, the corresponding complex local system is semisimple). This assumption is used in two places later: it ensures that the associated point in Betti moduli lies in the semisimple locus where finiteness statements are most naturally formulated, and it allows us to invoke finiteness principles of Deligne–Simpson type which are stated for integral variations of Hodge structure with semisimple underlying local system.
Finally, we clarify what is meant by ``control of denominators’’ for motivated projectors, since this is the bridge to the integrality condition on formal isomonodromy to be introduced next. A correspondence p ∈ CHd(Y×UY) ⊗ Q may require denominators, and even if Y and U are defined over R, there is no reason that pdR should be defined over R as an endomorphism in MIC(U/R) without inverting finitely many elements. In practice, one chooses a single N ∈ R \ {0} clearing the denominators of the cycle class, so that after replacing R by R[1/N] the correspondence is represented by an actual cycle with coefficients in R[1/N], and its de Rham realization produces an endomorphism of the Gauss–Manin system defined over R[1/N]. This process is compatible with restriction to fibers and with base change, and it ensures that reductions modulo primes of R[1/N] make sense for the projector away from a finite set of primes.
This spreading-out property is the precise reason we work with correspondences rather than arbitrary horizontal idempotents: it is the input that permits a uniform choice of N controlling all arithmetic realizations simultaneously. In the next section we will impose a parallel integrality condition on the horizontal deformation in de Rham moduli. The eventual argument will combine these two integrality statements—integrality of the formal isomonodromy leaf and integrality of the motivated projector—to propagate filtrations from the ambient Picard–Fuchs object to the summand (E, ∇) along the formal leaf, and thereby to deduce algebraicity of the corresponding isomonodromy locus.
We now make precise the integrality condition imposed on the formal isomonodromic deformation of a point in the relative de Rham moduli. Conceptually, this is a condition on the formal section of MdR(X/S, r) determined by isomonodromy: we require that this formal section, a priori defined over the fraction field K, in fact be defined over ŜR, s after inverting a single element of R. This is the arithmetic input that will allow us, in the next step, to compare characteristic 0 and characteristic p filtrations along the formal leaf.
Let $\widehat{\mathcal{O}}_{S,s}$ be
the 𝔪s-adic
completion of the local ring 𝒪S, s, and
write
$$
\widehat{S}_{R,s}:=\mathrm{Spf}(\widehat{\mathcal{O}}_{S,s}),\qquad
\widehat{S}_{K,s}:=\widehat{S}_{R,s}\,\widehat{\otimes}_R
K=\mathrm{Spf}(\widehat{\mathcal{O}}_{S,s}\otimes_R K).
$$
(Here $\widehat{\otimes}$ is harmless
since $\widehat{\mathcal{O}}_{S,s}$ is
already complete and K is a
localization of R.) We will
view ŜK, s
as the formal neighborhood of sK inside SK.
By construction, the stack MdR(X/S, r) is
defined over R, and it carries
its isomonodromy (horizontal) structure, which may be packaged as a
crystal on the smooth locus over S (equivalently, as a flat
Ehresmann-type connection on the moduli problem). For our purposes we
only use the following consequence: given a K-point
[(Xs, E, ∇)] ∈ MdR(X/S, r)(K),
there is a canonically associated through this point, i.e. a morphism of
formal stacks over SK
ℓ̂K : ŜK, s → MdR(X/S, r)/SK
which sends the closed point sK to [(Xs, E, ∇)]
and is horizontal for the isomonodromy crystal. We refer to ℓ̂K as the
through [(Xs, E, ∇)].
Pulling back the universal family on MdR(X/S, r) (on an
appropriate atlas if necessary) yields a formal family of flat bundles
on the pullback of X to ŜK, s.
Concretely, if we write
XŜK, s := X×SŜK, s,
then ℓ̂K
determines a flat bundle
$$
(\widehat{E},\widehat{\nabla})\in
\mathrm{MIC}(X_{\widehat{S}_{K,s}}/\widehat{S}_{K,s})
$$
whose restriction to the special fiber over sK identifies
with (E, ∇)K under
the canonical identification XsK ≃ Xs⊗RK.
The defining property of $(\widehat{E},\widehat{\nabla})$ is that it
is the formal deformation obtained by transporting $(E,)_K horizontally
along the crystal; in other words, $(\widehat{E},\widehat{\nabla})$ is the
restriction of the universal isomonodromy crystal to ŜK, s.
The integrality condition asserts that the preceding formal deformation, although defined over K, is induced by a formal deformation defined over R away from finitely many primes.
Several remarks clarify the meaning and the strength of this condition.
Since MdR(X/S, r) is
defined over R, to give $(\widehat{E}_{R},\widehat{\nabla}_{R})$ as
above is equivalent to giving a morphism
ℓ̂R : ŜR, s[1/N] → MdR(X/S, r)
whose base change to K
recovers ℓ̂K. Thus
integrality is a descent property of the itself. In particular, it is
independent of auxiliary presentations of MdR and stable under replacing MdR by an 'etale atlas.
After shrinking S around s we may assume S = Spec(A) is affine and s corresponds to a maximal ideal 𝔪 ⊂ A. Then ŜR, s is Spf(Â𝔪), and the deformation $(\widehat{E},\widehat{\nabla})$ may be described by a vector bundle Ê on X×SSpf(Â𝔪⊗RK) together with an integrable connection relative to Â𝔪⊗RK. Descent to ŜR, s[1/N] means that, after choosing trivializations on a Zariski cover, the corresponding connection matrices can be chosen with coefficients in Â𝔪[1/N] rather than merely in Â𝔪⊗RK. Equivalently, the formal Taylor coefficients governing the horizontal transport on moduli have no denominators away from N.
If a descent exists, it is unique up to unique isomorphism. Indeed, base change from ŜR, s[1/N] to ŜK, s is faithful on vector bundles with integrable connection (one may reduce to the corresponding statement for finite projective modules over a domain and use that K is the total fraction field). Thus the integrality condition is a property, not additional structure.
The moduli stack and the connection (E, ∇) may have good reduction at almost all primes of R but fail at finitely many. Requiring descent over ŜR, s without inverting anything would exclude many natural geometric situations. The formulation with R[1/N] matches the usual paradigm of ``spreading out away from finitely many primes’’ and is exactly what is needed to perform reduction modulo 𝔭 for all primes 𝔭 ∤ N.
It is useful to isolate a more local valuation-theoretic version of the same condition, which clarifies the analogy with classical integrality constraints for linear differential equations.
Let 𝔭 ⊂ R[1/N] be
a height-one prime of a normal finite type R[1/N]-algebra dominating
the local situation, so that the localization R[1/N]𝔭 is a
discrete valuation ring. Fix a uniformizer ω = ω(𝔭). We may form the
ω-adic completion R̂𝔭, and the corresponding
base change of the formal neighborhood,
$$
\widehat{S}_{\widehat{R}_{\mathfrak{p}},s}:=\widehat{S}_{R,s}\,\widehat{\otimes}_{R[1/N]}
\widehat{R}_{\mathfrak{p}}.
$$
Then the deformation $(\widehat{E},\widehat{\nabla})$ yields by
base change a formal flat bundle over the fraction field of R̂𝔭, and we may ask
whether it is defined over R̂𝔭 itself.
The global descent to ŜR, s[1/N] implies ω-integrality for every such 𝔭 ∤ N, and conversely one may view the global descent condition as asserting ω-integrality uniformly at all height-one primes away from N.
This viewpoint makes transparent that the condition is not merely that (E, ∇) itself admit an integral model on the special fiber, but rather that the defining the isomonodromy leaf (i.e. the horizontal section of MdR) admit an integral formal solution. In particular, it controls denominators of the entire formal leaf, not just the initial point.
We now explain how this integrality condition should be compared with the classical Grothendieck–Katz p-curvature conjecture and with more familiar integrality notions for linear connections.
Let (V, ∇V) ∈ MIC(U/K)
be a vector bundle with integrable connection on a smooth K-scheme U. To speak about its p-curvature at a prime p, one first needs an integral model
over R[1/N] so that
one can reduce modulo primes 𝔭 ⊂ R[1/N] of residue
characteristic p. When such a
model exists, one obtains for almost all 𝔭 a reduced connection (V𝔭, ∇𝔭) on
U𝔭, hence a p-curvature operator
ψp(∇𝔭) : 𝒯U𝔭/κ(𝔭) → End(V𝔭)
measuring the failure of ∇𝔭
to be compatible with the pth
power operation on derivations. The Grothendieck–Katz conjecture
predicts that if ψp(∇𝔭) = 0
for almost all p, then the
complex local system associated to (V, ∇V) has
finite monodromy.
Two features of this picture are relevant for us.
The very definition of ψp requires that (V, ∇V) extend over an integral model. Likewise, our later arguments will require reducing along the formal leaf modulo primes. The descent of $(\widehat{E},\widehat{\nabla})$ to ŜR, s[1/N] is precisely the condition that allows such reductions . Without it, one could perhaps reduce the initial flat bundle (E, ∇), but there would be no reason that the entire formal horizontal transport on moduli be defined modulo 𝔭.
The isomonodromy foliation on MdR(X/S, r) is a nonlinear flat connection on a moduli problem. Asking that its formal horizontal section through a K-point be integral is, in this sense, a nonlinear integrality constraint comparable to asking that a linear differential equation admit integral solutions (or that its Taylor coefficients be integral). In the linear setting, conditions on p-curvature are expected to force strong arithmetic and geometric finiteness properties; in our nonlinear setting, the conclusion we seek is the algebraicity of the isomonodromy leaf (equivalently, finiteness of the induced orbit in Betti moduli). The role of integrality is analogous: it supplies the possibility of comparing characteristic 0 and characteristic p structures along the deformation, which in turn constrains the complexity of the leaf.
In particular, for Gauss–Manin connections one knows that, modulo almost all primes, the relevant p-curvature is nilpotent (indeed, it is controlled by Frobenius and the Hodge-to-de Rham spectral sequence). The Lam–Litt theorem that we will use next can be viewed as an implementation of this philosophy in the isomonodromy setting: integrality of the formal isomonodromic deformation is established for Picard–Fuchs objects by exploiting precisely the additional structures available in characteristic p (notably the conjugate filtration and Frobenius). In the present work we isolate the output needed for our argument as a black box: the descent of the formal leaf, one obtains extension of filtrations along the leaf.
It is worth emphasizing that the Grothendieck–Katz conjecture concerns a connection (V, ∇V), whereas our condition concerns a obtained by horizontal transport in moduli. Even if (E, ∇) on Xs extends over R[1/N] as a connection on that fixed fiber, this says nothing about the behavior of its isomonodromic deformations: denominators can (and typically do) appear in the deformation parameters when one solves the corresponding nonlinear differential equation formally. The descent to ŜR, s[1/N] rules out precisely this phenomenon.
Assuming integrality, for each prime 𝔭 ∤ N of R[1/N] one may reduce the entire formal deformation modulo 𝔭. More precisely, after base change to the 𝔭-adic completion of $\widehat{\mathcal{O}}_{S,s}[1/N]$ and then reduction to the residue field κ(𝔭), we obtain a formal flat bundle in characteristic p on the corresponding special fiber of X, equipped with the property that it remains horizontal for the reduced isomonodromy crystal. This supplies a well-defined setting in which to compare:
This comparison is the bridge between arithmetic input and geometric output. In the next section we will invoke the Lam–Litt theorem, which precisely asserts that in the Picard–Fuchs case, integrality of the formal isomonodromy deformation forces such filtration-extension phenomena. Combined with the functoriality for motivated projectors established earlier, we will then transport the filtration extension from the ambient Gauss–Manin system to the motivated summand (E, ∇), and from there deduce the desired finiteness of the Betti orbit (hence algebraicity of the isomonodromy leaf).
Finally, we note that the integrality condition is stable under inverting additional elements of R and under shrinking S around s. Thus, throughout the argument, we will freely replace R by R[1/N′] for multiples N′ of N and replace S by a Zariski-open neighborhood of s without further comment; the content of the hypothesis is that such N exists for which the horizontal formal deformation is defined over ŜR, s[1/N] rather than only over ŜK, s.
In the subsequent arguments we will require a mechanism which turns the arithmetic hypothesis of integrality of the formal isomonodromy leaf into geometric constraints on the corresponding formal deformation. In the Picard–Fuchs situation, namely for Gauss–Manin connections, such a mechanism is provided by a theorem of Lam–Litt. We record here a black-box formulation tailored to our later use.
Let π : Y → U be
smooth and proper over R, and
fix i ≥ 0. Write
ℋ := (Riπ*ΩY/U•, ∇GM) ∈ MIC(U/R)
for the Gauss–Manin connection. After possibly shrinking U and inverting a single element of
R, we may and do assume that
the Hodge-to-de Rham spectral sequence for π degenerates at E1 and that each Rbπ*ΩY/Ua
is locally free, so that the
Faℋ := Im(Riπ*ΩY/U ≥ a → Riπ*ΩY/U•)
is a filtration by subbundles on U. It is Griffiths-transverse with
respect to ∇GM, i.e.
∇GM(Faℋ) ⊆ Fa − 1ℋ ⊗ ΩU/R1.
For primes of good reduction one also has the . More precisely, let
𝔭 be a prime of R[1/N] of residue
characteristic p > 0, and
base change π to κ(𝔭):
π𝔭 : Y𝔭 → U𝔭.
Then the relative Frobenius and the Cartier isomorphism produce an
increasing filtration C• on the de Rham
cohomology bundle
ℋ𝔭 := Ri(π𝔭)*ΩY𝔭/U𝔭•
whose associated graded pieces are identified (non-canonically at the
level of total objects, but canonically on graded pieces) with Frobenius
pullbacks of coherent cohomology:
graC(ℋ𝔭) ≃ FU𝔭*(Ri − a(π𝔭)*ΩY𝔭/U𝔭a),
and C• is
horizontal for the induced connection (in the sense appropriate to
characteristic p, e.g. for the
associated stratification/crystal). We emphasize that F• and C• are constructions for
Gauss–Manin objects; the Lam–Litt input is that, under isomonodromy and
an integrality hypothesis, these filtrations persist along the formal
deformation even though the deformed flat bundle is not assumed to come
from geometry.
We now place ourselves in the isomonodromy situation. Let ŜR, s
be the formal completion of S
at s, and suppose we are given
an integral formal leaf after inverting N, i.e. a descended formal flat
bundle
$$
(\widehat{V}_R,\widehat{\nabla}_R)\in
\mathrm{MIC}\bigl(X_{\widehat{S}_{R,s}[1/N]}/\widehat{S}_{R,s}[1/N]\bigr)
$$
whose base change to ŜK, s
is the formal isomonodromic deformation of the given K-point. In applications we will
take V to be (the restriction
to a suitable open of a fiber of) a Gauss–Manin system ℋ, but for the present discussion it is
convenient to isolate the formal notion of a filtration along the
leaf.
In the Gauss–Manin case, F0• will be the Hodge filtration on the special fiber (and, after reduction modulo 𝔭, the relevant increasing filtration will be the conjugate filtration). The key point is that the of such extensions is highly nontrivial for an arbitrary isomonodromic deformation; it is precisely what Lam–Litt prove under integrality assumptions in the Picard–Fuchs setting.
We state a form of the result sufficient for our purposes. The precise formulation in is more elaborate (including compatibility with Frobenius, strong divisibility properties, and uniformity statements), but we will only use the existence of filtrations extending the special fiber and the fact that they are natural for horizontal morphisms.
We stress two aspects of Theorem~.
First, the conclusion is a statement , not only about the special fiber. Even if the special fiber Gauss–Manin connection has its Hodge filtration, there is no reason for that filtration to propagate along an isomonodromic deformation in moduli: the deformation is defined by solving a nonlinear horizontality condition, and one expects denominators and obstructions without an arithmetic input. The integrality hypothesis precisely supplies the control needed to force existence of F̂• and Ĉ•.
Second, for our later use, we will apply part~(c) in situations where the relevant horizontal morphisms arise from algebraic correspondences. Concretely, if p ∈ CHdim (Y/U)(Y×UY) ⊗ Q is a correspondence, its de Rham realization yields a horizontal endomorphism of ℋ; after spreading out away from finitely many primes, the same remains true along the descended formal deformation. Functoriality then implies that the extended filtrations are preserved by such endomorphisms (and, in particular, by the motivated projectors that will cut out our summands).
Both the construction of the Hodge filtration as a filtration by subbundles and the existence of the conjugate filtration require standard ``spreading out’’ and degeneration hypotheses, which we ensure by shrinking and inverting a single element. Theorem~ should therefore be read in the same sense as our integrality hypotheses: after replacing R by R[1/N′] and shrinking the neighborhood of s in S, the stated filtrations exist and are compatible with reduction at all primes away from N′.
In many of the situations treated by Lam–Litt, the filtration extension is rigid in the sense that it is uniquely determined by the requirement that it extends the special fiber and satisfies Griffiths transversality (together with a mild normalization condition). We will not require a separate uniqueness statement, but we implicitly use the rigidity built into functoriality: once an extension is constructed canonically, horizontal endomorphisms coming from geometry must preserve it.
Later, we will apply Theorem~ only through the following
implication:
integral formal isomonodromy for a
Gauss–Manin object ⇒ existence of an extended Griffiths-transverse
Hodge filtration
(and its characteristic p
companion Ĉ• along
reductions). No other input from Lam–Litt will be needed in our
argument.
This completes the Picard–Fuchs filtration-extension input. In the next step, we will explain how to pass these filtration-extension statements from an ambient Gauss–Manin system to a motivated semisimple direct summand, using integral control on idempotent projectors and compatibility of correspondences with Hodge and conjugate filtrations.
We now explain how to transfer the filtration-extension statement obtained for a Gauss–Manin object to a semisimple direct summand cut out by a idempotent correspondence. This is the point where we use more than abstract semisimplicity: we require an idempotent projector whose denominators are controlled uniformly (in characteristic 0 and modulo almost all primes), so that the extended filtrations produced by the Picard–Fuchs input remain stable under the projector.
Throughout this subsection we work over a sufficiently small neighborhood of s in S and after inverting a single element of R (which we enlarge as needed, without further comment). In particular, we assume that:Let ℋ = (Riπ*ΩY/U•, ∇GM) ∈ MIC(U/R)
and let
p ∈ CHdim (Y/U)(Y×UY) ⊗ Q
be an idempotent correspondence whose de Rham realization pdR is a horizontal
idempotent endomorphism of ℋ with image
isomorphic to (E, ∇)|U.
The first issue is that p is only given with Q-coefficients. In order to speak meaningfully about reduction modulo 𝔭 and about endomorphisms along a formal deformation defined over ŜR, s[1/N], we must bound denominators. Concretely, after replacing R by R[1/N] for some N, we may assume that p is represented by a cycle with coefficients in R[1/N] and that its de Rham realization is defined over R[1/N].
We also need to propagate the idempotent from the special fiber to the descended formal deformation. For this we use the fact that isomonodromy deformations are by construction: they carry a canonical stratification/crystal structure over the formal base, and horizontal endomorphisms are precisely the endomorphisms in the corresponding crystal category. In particular, once we have an idempotent in the special fiber that is horizontal, there is at most one way to extend it as a horizontal endomorphism along a fixed isomonodromic deformation. Existence comes from the motivic origin: p defines an endomorphism of the relative cohomology object, hence of its crystal realization, so it extends over the same base as soon as the deformation is defined over that base.
We package this as follows.
We next verify that pdR is for the filtrations we intend to extend. On the special fiber, this is a purely functorial statement: the Hodge filtration on Gauss–Manin de Rham cohomology is functorial for morphisms induced by correspondences, and the same holds for the conjugate filtration after reduction modulo 𝔭 (via functoriality of the Cartier operator and of Frobenius pullback on the graded pieces).
Thus, after inverting N as in Lemma~, we may assume:
We now combine this filteredness with the Lam–Litt functoriality statement for extended filtrations along the descended formal leaf. Namely, since p̂dR is horizontal, Lam–Litt’s naturality implies that p̂dR is strictly compatible with the extended Hodge filtration F̂• (and similarly with Ĉ• after reduction). We record the consequence we will actually use.
We now isolate the purely formal mechanism which passes a Griffiths-transverse filtration to the image of a filtered horizontal idempotent. This is the key ``summand-stability’’ statement; the motivic input above serves only to verify its hypotheses in our situation.
The same formal argument applies to an filtration (e.g. Ĉ• in characteristic p), and we will use it in that form as well.
We return to our geometric situation. Let $(\widehat{\mathcal{H}}_{R},\widehat{\nabla}_{R})$
be the descended isomonodromic deformation of ℋ over ŜR, s[1/N]
for which Lam–Litt supplies the extended filtrations F̂• (and Ĉ• modulo 𝔭 ∤ N). Let p̂dR be the horizontal
idempotent on $\widehat{\mathcal{H}}_{R}$ given by Lemma~,
which preserves these filtrations by Lemma~. We then set
$$
(\widehat{E}_{R},\widehat{\nabla}_{R})
\;:=\;
\bigl(\mathrm{Im}(\widehat{p}_{\mathrm{dR}}),\,\widehat{\nabla}_{R}|_{\mathrm{Im}(\widehat{p}_{\mathrm{dR}})}\bigr).
$$
By Lemma~, the filtrations F̂• and (after reduction)
Ĉ• restrict to
filtrations on $(\widehat{E}_{R},\widehat{\nabla}_{R})$
extending the induced filtrations on the special fiber. On that special
fiber, the induced filtration is precisely the Hodge filtration on the
Picard–Fuchs summand (E, ∇)|U ≃ Im(pdR),
because pdR acts by
filtered endomorphisms and hence FaE = pdR(Faℋ).
Two remarks are in order.
By construction, $(\widehat{E}_{R},\widehat{\nabla}_{R})$ is a formal flat bundle deforming (E, ∇)|U and obtained by applying a horizontal projector to an isomonodromic deformation of ℋ. In particular it is itself isomonodromic (its underlying Betti local system is the corresponding direct summand of the Gauss–Manin local system). Hence, after base change to K, it identifies with the formal isomonodromic deformation of (E, ∇)K along ŜK, s; uniqueness of isomonodromic deformation in the crystal formalism gives the identification.
The argument above requires that p̂dR exists over the same integral base as the descended deformation and that it remains idempotent modulo 𝔭 away from N. Without this control, one can still split after extending scalars to K, but the resulting summand need not descend to ŜR, s[1/N], and the filtration-extension statement cannot be transported to arithmetic information modulo 𝔭. Thus the motivated-projector hypothesis is exactly what allows the Lam–Litt filtrations to be inherited by the summand .
Finally, we indicate the Tannakian upgrade used later: once filtration extension is known for Gauss–Manin generators, it persists under the Tannakian operations, and the only genuinely new input for semisimple subquotients is that the corresponding idempotents admit integral motivated realizations.
Let ⟨PF⟩ denote the smallest neutral Tannakian subcategory of MIC(U/K) generated by Gauss–Manin systems and closed under semisimple subquotients. Suppose (V, ∇) ∈ ⟨PF⟩ is semisimple and that, after inverting a single N, each simple factor of V is cut out inside some tensor construction on Gauss–Manin objects by an idempotent coming from a motivated correspondence (equivalently: the relevant idempotent in the Tannakian category is realized by a horizontal endomorphism defined over R[1/N] and spreading modulo 𝔭 ∤ N).
Then the previous discussion applies inductively:In this way one reduces the extension problem for (V, ∇) to the Gauss–Manin case treated by Lam–Litt, together with the integrality and filteredness of the relevant motivated projectors.
We will use this principle in the next step only through the following output: This is the precise bridge from the arithmetic integrality hypothesis to the geometric finiteness mechanism invoked in the subsequent section.
We now explain how the filtration-extension output obtained in the previous subsection forces finiteness of the π1(S)-orbit of the associated Betti point. We work after fixing an embedding K ↪ C, and we implicitly replace S by a sufficiently small connected Zariski-open neighborhood of s (so that analytification is well-behaved and the relevant moduli stacks may be treated as analytic orbifolds on their smooth loci).
Let U ⊆ Xs
be the dense open on which (E, ∇)|U is
realized as a motivated semisimple summand of a Gauss–Manin system. Via
the chosen embedding into C, we obtain a complex local
system
𝕍C := ker (∇an : Ean → Ean ⊗ ΩU(C)an1),
and, by semisimplicity, a reductive Betti point
[ρ] ∈ MB(U, r)(C).
Because (E, ∇)|U is a
cohomological summand cut out by a motivated projector, it is equipped
(after inverting a single integer, which we suppress in the notation)
with a Z-structure and
a polarization coming from the ambient cohomology and Poincar'e duality.
Concretely, we may and do choose a Z-local system 𝕍Z on U(C)an
and an identification 𝕍Z ⊗ C ≃ 𝕍C
compatible with the splitting induced by the idempotent correspondence,
so that the corresponding holomorphic flat bundle is our (E, ∇)|UC.
We consider the locus in Betti moduli consisting of those points
which underlie a polarizable Z-variation of Hodge
structure on U of the same
rank and with the same Hodge numbers as our summand (equivalently, the
same Hodge filtration ranks as the induced filtration on E). We denote this subset by
MB(U, r)VHSZ ⊆ MB(U, r).
Two structural facts are used in the sequel.
By non-abelian Hodge theory, the analytification MB(U, r)an is
analytically isomorphic to the Dolbeault moduli space MDol(U, r)an
of polystable Higgs bundles with vanishing Chern classes. The C×-action on
MDol by scaling the Higgs
field has fixed-point locus equal to the set of complex variations of
Hodge structure (more precisely, complex VHS correspond to those
polystable Higgs bundles admitting a Hodge decomposition, which is
equivalent to being fixed under the C×-action). Since
the fixed-point locus of an algebraic C×-action is a
closed algebraic subspace, it follows that the complex-VHS locus in
MB(U, r) is
algebraic. Imposing the existence of an integral lattice compatible with
the Hodge filtration amounts to a further arithmetic condition on the
corresponding local system; for our purposes it suffices that
MB(U, r)VHSZ(C)
is a union of connected components of the complex-VHS locus intersected
with the (countable) set of Z-local systems. In
particular, it is stable under algebraic automorphisms of MB(U, r) coming from outer
automorphisms of π1(U).
This is the sense in which Simpson’s ``non-abelian Noether–Lefschetz loci’’ enter: loci defined by the existence of additional Hodge-theoretic structure on a local system are Zariski-closed (or, in families, a countable union of closed algebraic subsets), and hence provide algebraic constraints on any set of representations characterized by such structure.
Deligne’s finiteness theorem asserts, in the form relevant here, that
for a fixed smooth quasi-projective complex variety U and a fixed rank r, there are only finitely many
isomorphism classes of irreducible polarizable Z-variations of Hodge
structure of rank r with
prescribed local behavior at infinity (quasi-unipotent local monodromy)
and prescribed Hodge numbers.
In particular, the set of points
MB(U, r)VHSZ(C)
with fixed Hodge numbers is finite modulo isomorphism of local systems,
hence corresponds to a finite subset of the set of C-points of the character
variety/stack.
We emphasize that the finiteness statement is genuinely about VHS; without integrality, the complex-VHS locus can have positive dimension, so no finiteness is possible at the level of C-local systems.
We now use the filtration-extension statement along the integral formal isomonodromic deformation obtained previously for our summand (E, ∇)|U.
Let ŜC, s
be the formal completion of S(C)an
at s, and let
$$
(\widehat{E},\widehat{\nabla},\widehat{F}^\bullet)
$$
denote the formal isomonodromic deformation of (E, ∇)|UC
together with the extended Griffiths-transverse filtration F̂• (the extension
produced by combining the Picard–Fuchs input with summand-stability).
The crucial point is that this filtration is not an auxiliary
decoration: it is functorial for horizontal morphisms in the
isomonodromy crystal category. Consequently, whenever we analytically
continue the isomonodromy germ around a loop in S(C)an,
the transported germ carries a transported filtration of the same
type.
More precisely, let γ ∈ π1(S(C)an, s). The outer monodromy of the family X → S along γ yields an outer automorphism of π1(U(C)an) (after choosing basepoints and identifying fibers along γ), hence an induced algebraic automorphism of MB(U, r). The point γ ⋅ [ρ] is, by definition, the value at s of the analytic continuation of the (multi-valued) horizontal section of the relative Betti moduli determined by ρ. Under the Riemann–Hilbert correspondence, the same continuation acts on the de Rham point and yields a new flat bundle on UC with monodromy representation γ ⋅ ρ.
Because F̂• is defined as part of the horizontal structure along the isomonodromy deformation and is functorial for horizontal maps, the analytically continued object at the end of the loop again carries a Griffiths-transverse filtration of the same Hodge type. Moreover, the integral structure (the Z-lattice) is transported as well: the isomonodromic deformation is, on the Betti side, literally constant as a local system on the of the pulled-back family along the loop, and therefore carries the same Z-local system throughout. Evaluating at the endpoint s yields a Z-local system on U with underlying complex local system γ ⋅ 𝕍C.
We therefore obtain:
Equivalently, the entire π1(S)-orbit of [ρ] is contained in MB(U, r)VHSZ(C) (for the fixed Hodge numbers determined at s).
This containment is the point where the formal filtration extension is used: without an extension along the isomonodromic germ, there is no reason for analytic continuation of the de Rham point to return with a compatible Hodge filtration, even if the original point happened to be of Hodge type.
We may now combine the previous two inputs.
Since all γ ⋅ [ρ]
lie in MB(U, r)VHSZ(C)
with fixed Hodge numbers, Deligne’s finiteness theorem implies that the
set
{γ ⋅ [ρ] ∣ γ ∈ π1(S(C)an, s)}
is finite. In other words, the π1(S(C)an, s)-orbit
of the Betti representation ρ
in MB(U, r) is
finite.
It is convenient to rephrase this conclusion in a way that exhibits
Simpson’s algebraicity result as well. Let
NLZna ⊆ MB(U, r)
denote the (non-abelian) Noether–Lefschetz locus of integral VHS points
of the relevant Hodge type. Simpson’s theorem shows that the complex-VHS
locus is Zariski-closed, hence NLZna
is contained in a countable union of closed algebraic subsets; Deligne’s
finiteness shows that its set of C-points of the fixed
integral type is finite. The previous subsection shows that our orbit is
contained in this locus. Thus the orbit is not merely countable (which
is automatic), but forced to be finite by the arithmetic rigidity of
Z-VHS.
We will use this finiteness statement as the Betti-side input in the final step: via the identification between finite π1(S)-orbits on MB(Xs, r) and algebraic leaves of the isomonodromy foliation, finiteness of the orbit is equivalent to algebraicity of the isomonodromy leaf through [(Xs, E, ∇)].
We record several situations in which the hypotheses of Theorem~A (and its Tannakian variant Theorem~B) are naturally satisfied, and we indicate how the conclusion translates into concrete finiteness statements on character varieties. Throughout we work in the global setup fixed in the introduction: a smooth proper family f : X → S over a finitely generated integral Z-algebra R, a point s ∈ S(R) with fiber Xs, and a rank r flat bundle (E, ∇) on Xs/R which is, on a dense open U ⊆ Xs, a semisimple motivated direct summand of a Gauss–Manin system. We also assume the integrality of the formal isomonodromic deformation after inverting a single N ∈ R.
Assume for the moment that f : X → S is a
family of smooth projective curves of genus g ≥ 0 (or, more generally, a family
of pointed curves in a logarithmic variant). Fix an embedding K ↪ C. The
associated outer action of π1(S(C)an, s)
on π1(Xs(C)an)
factors through a monodromy subgroup of the mapping class group Γg, n
(after choosing markings and allowing punctures if present). Thus a
Betti point
[ρ] ∈ MB(Xs, r)(C)
gives rise to an orbit under a concrete discrete group, namely the image
of π1(S)
in Out(π1(Xs)).
In this setting, the conclusion of Theorem~A may be read as a ``finite mapping class orbit’’ statement: under our motivic Picard–Fuchs summand hypothesis and integral formal isomonodromy, the orbit of [ρ] under the monodromy subgroup of Γg, n is finite. This is precisely the content of Corollary~C in the curve case, but we emphasize that no explicit description of the orbit is required: finiteness is forced purely by the existence of an integral isomonodromic deformation together with geometric origin (in the summand sense) and semisimplicity.
It is useful to compare this to the classical theory of isomonodromy for punctured P1. There, algebraic solutions of Painlev'e-type equations are often characterized by finiteness of the braid group orbit on the relevant character variety. In the present higher genus setting, Γg, n plays the role of the braid group, and the isomonodromy leaf in MdR(X/S, r) is the algebro-geometric avatar of the isomonodromic deformation. Theorem~A supplies a general mechanism for proving finiteness of mapping class group orbits when integrality (in the formal sense) can be verified.
One common source of such integrality is the Picard–Fuchs situation treated by Lam–Litt: if (E, ∇)|U is a Gauss–Manin system itself, then integral formal isomonodromy is automatic away from finitely many primes of R. The contribution of the summand-stability results is that we may pass from the ambient Gauss–Manin system to motivated semisimple direct summands cut out by correspondences without losing the integrality input.
Let a : A → U be an
abelian scheme over a smooth R-scheme U, and consider the Gauss–Manin
connection
ℋA := (R1a*ΩA/U•, ∇GM) ∈ MIC(U/R).
If A carries additional
endomorphisms (for instance real multiplication or quaternionic
multiplication), then the algebra End0(A/U) := End(A/U) ⊗ Q
acts by correspondences on A
and hence on ℋA.
Choosing an idempotent p ∈ End0(A/U)
yields a motivated projector in the sense of our hypotheses: the
correspondence defined by the graph of p cuts out a direct summand of
cohomology. After inverting a single integer to clear denominators, such
projectors are defined over R[1/N] and induce
horizontal idempotents on ℋA.
In particular, if (E, ∇)|U is realized as Im(pdR) for such an idempotent, then (E, ∇)|U is a Picard–Fuchs summand with motivated projector. Semisimplicity is automatic on the Betti side for polarizable variations of Hodge structure, hence for weight-one factors arising from abelian schemes. Moreover, the integral formal isomonodromy input is available for ℋA by Lam–Litt, and therefore for the summand by our filtration-compatibility and summand-stability lemmas.
A concrete instance that motivates this discussion is the ``GL2-type’’ situation: suppose a rank-2 summand of R1a*Q is cut out by an idempotent in a totally real field of endomorphisms. The associated rank-2 local system on U(C)an is then geometric, polarizable, and often exhibits arithmetic constraints that are difficult to detect purely at the level of character varieties. Theorem~A implies that, for any family X → S in which such a summand appears on a fiber and whose formal isomonodromy is integral, the corresponding Betti point has finite π1(S)-orbit. From the character variety viewpoint, this produces a supply of rigid (in the dynamical sense) points whose rigidity is certified by motivic projectors and arithmetic integrality rather than by explicit monodromy computations.
We next describe a class of examples where the motivated projector
hypothesis can be verified by elementary means, without appealing to
deep statements about motivated cycles. Let U parametrize a family of cyclic
covers of P1, for example
affine curves of the form
$$
y^m = \prod_{j=1}^n (x-t_j)
$$
with the branch points (t1, …, tn) ∈ U,
and let π : Y → U be a
smooth proper model of the family. The deck group G ≃ Z/mZ
acts on Y over U, hence on the Gauss–Manin system
ℋ = (Riπ*ΩY/U•, ∇GM).
For each character χ : G → C×,
the group algebra idempotent
$$
e_\chi \;=\; \frac{1}{|G|}\sum_{g\in G}\chi(g)^{-1}g\in
\mathbf{Q}(\mu_m)[G]
$$
defines a projector onto the χ-isotypic component. This yields a
direct sum decomposition of ℋ after
extending scalars, and in particular produces explicit idempotent
correspondences on Y×UY
coming from the G-action.
In this setting, the ``motivated’’ nature of the idempotents is tautological: they are induced by algebraic correspondences (graphs of automorphisms) and hence lie in the relevant Chow groups with controlled denominators. The only denominators are those of |G|, so one may take N = m (or a divisor of m depending on the precise idempotent). Therefore the hypotheses of Theorem~A are readily checkable.
The conclusion has a particularly concrete interpretation in rank r = 2 cases where an isotypic factor has small rank and gives rise to a non-abelian character variety of manageable dimension. One then obtains finiteness of the orbit under the natural monodromy group acting on the character variety of the fiber curve (or punctured curve). This recovers, in a uniform way, finiteness phenomena that in classical treatments are established by explicit braid group computations: the argument here does not require computing the orbit, only verifying the geometric summand condition and integral isomonodromy.
Katz’s middle convolution operation produces many rigid and cohomologically rigid local systems on P1 \ {t1, …, tn} starting from rank-one data and iterating MCλ for suitable parameters λ. In favorable cases, the resulting local systems admit geometric realizations as direct factors of cohomology of families of covers (often after introducing auxiliary variables and compactifying). When such a realization is available over R, the resulting de Rham object is again a Picard–Fuchs summand in our sense.
More precisely, suppose a rank-r local system 𝕍 on U is known to be isomorphic (after semisimplification) to a direct summand of Riπ*Q for some smooth proper π : Y → U, and assume that the summand is cut out by an algebraic correspondence (for instance coming from a group action or from an explicit projector in an endomorphism algebra). Then (E, ∇) associated to 𝕍 satisfies hypothesis~(1). If, moreover, (E, ∇) occurs in a family with integral formal isomonodromy, Theorem~A forces finiteness of the corresponding orbit on the relevant character variety.
In genus 0 this often matches the expected behavior of isomonodromy equations: rigid local systems (in Katz’s sense) tend to yield algebraic isomonodromy solutions. Our point is that the mechanism is robust under passing to motivated summands and does not rely on rigidity in the deformation-theoretic sense. In particular, one can have positive-dimensional moduli of representations while still obtaining finite orbits for certain arithmetic-geometric points singled out by integral isomonodromy.
We conclude this section by isolating which parts of the argument genuinely use the strengthened ``motivated projector’’ hypothesis, and where one might hope to weaken assumptions.
Semisimplicity is used at two distinct points. First, it ensures that the Betti point lies in the reductive locus of MB so that the non-abelian Hodge correspondence is well-behaved and the orbit problem is naturally formulated on character varieties. Second, semisimplicity allows us to treat (E, ∇)|U as a direct summand rather than merely a subobject in MIC, which is essential for applying idempotent projectors and for passing filtrations along isomonodromy in a controlled way. In geometric situations (polarizable variations of Hodge structure, or direct factors of Gauss–Manin systems), semisimplicity is typically automatic over C, but may require care over general bases.
Even if (E, ∇)|U is known to be a semisimple direct summand of a Gauss–Manin system in MIC(U/K), the idempotent defining the summand a priori lives only over K and may involve uncontrolled denominators. Our proof requires that, after inverting a N ∈ R, the projector spreads out and remains compatible with both the Hodge filtration in characteristic 0 and the conjugate filtration mod p for almost all primes. This is exactly the content of Lemmas~1 and~2.
The motivated correspondence hypothesis is a convenient sufficient condition: correspondences have integral models away from finitely many primes, and their functoriality on filtrations is built into the formalism of de Rham and crystalline realizations. Without such control, filtration extension for the ambient Gauss–Manin system need not pass to the summand: one can construct idempotents in endomorphism algebras of flat bundles whose denominators force obstructions mod p, preventing any uniform integral structure. Thus the hypothesis is not merely a technicality; it is the place where arithmetic input enters the non-linear problem.
Our integrality assumption is formal and local on S at s, but it is global in the arithmetic direction: it asserts the existence of an R[1/N]-model of the formal isomonodromic deformation. For Picard–Fuchs systems this integrality is supplied by Lam–Litt, and in the examples above it can often be verified by reducing to such systems and invoking summand-stability. In practice, checking integrality beyond the Gauss–Manin world remains subtle and is best viewed as a non-linear analogue of p-curvature constraints.
In many of the explicit examples (cyclic covers, group actions, endomorphism algebras of abelian schemes) the relevant projectors are defined by honest algebraic endomorphisms with small denominators, so the motivated projector hypothesis is essentially automatic. One may then view Theorem~A as a general finiteness principle: a representation arises from a controlled cohomological construction and satisfies integral isomonodromy, it cannot wander in character varieties under base monodromy.
In Section~ we will return to the question of whether the motivated projector hypothesis can be removed or replaced by a purely Tannakian integrality condition, and to what extent the argument extends from direct summands to semisimple subquotients generated by Picard–Fuchs objects.
We conclude by indicating several natural directions in which one may hope to strengthen the main finiteness principle proved above. The common theme is that Theorem~A extracts algebraicity of isomonodromy leaves from two inputs: an input (integrality of the formal isomonodromic deformation) and a input (realization as a semisimple motivated direct summand of a Picard–Fuchs connection). The examples of Section~ show that both inputs occur widely, but they also highlight the two evident limitations of our method: (i) our reliance on correspondences to control denominators of idempotents, and (ii) our insistence on direct summands rather than more general semisimple subquotients. We also comment on possible generalizations from points/leaves to higher-dimensional invariant subvarieties, and on the shape of a potential ``full’’ nonlinear p-curvature conjecture.
In the proof of Theorem~A, the motivated-projector hypothesis enters through two concrete mechanisms: it provides an idempotent endomorphism defined over a single localization R[1/N] (Lemma~1), and it ensures functoriality of both the Hodge and conjugate filtrations with respect to that idempotent (Lemma~2). Once those two facts are available, the passage of filtration extension from the ambient Gauss–Manin object to the summand is purely formal (Lemma~3). Thus, if one seeks to remove the motivic hypothesis, one must replace it with a hypothesis that supplies these two properties.
A first possible weakening is to replace
motivated correspondence'' byabsolute Hodge (or absolute
Tate) projector’’ in the sense that the idempotent is required to be
compatible with all realizations. Concretely, one could ask that the
idempotent cutting out (E, ∇)|U inside
the chosen Picard–Fuchs object ℋ be
represented by an element of EndMIC(U/K)(ℋK)
which, after inverting a single N, extends to EndMIC(U/R[1/N])(ℋ)
and whose reductions mod 𝔭 preserve the
conjugate filtration for almost all primes 𝔭. This formulation is purely in terms of the
de Rham/crystalline categories and avoids explicit mention of Chow
groups; however, it is not evident how to verify it in geometric
situations without returning to correspondences.
A second, more conceptual approach is to express the needed ``denominator control’’ Tannakianly. Let ⟨ℋ⟩⊗ denote the neutral Tannakian subcategory of MIC(U/K) generated by ℋ under tensor operations, duals, and subquotients. The summand (E, ∇)|U corresponds to an idempotent in the coordinate Hopf algebra of the Tannaka group Gℋ. We may ask whether integrality of isomonodromy for ℋ, together with integrality properties of Gℋ (for instance, existence of a reductive group scheme model over R[1/N]), forces the idempotent to be integral after inverting the same N. In other words, one would like a criterion that turns an rational representation-theoretic projector into an integral one, uniformly in families.
Even in linear representation theory, this is delicate: an idempotent in EndK(V) need not preserve any fixed lattice in V unless one has strong control on denominators. In our context the relevant lattice is not merely a module over R[1/N], but a lattice compatible with connection and (crucially) compatible with the Hodge/conjugate filtrations across characteristics. One might therefore seek an ``integral Tannakian’’ condition saying that (E, ∇)|U is cut out by an idempotent in an endomorphism algebra that is already defined over the corresponding R[1/N]-Tannakian category of filtered objects produced by Lam–Litt along the integral isomonodromy leaf. If such an integral Tannakian condition were available, the motivic hypothesis could be replaced by a statement that the summand is defined by a idempotent in that category, which is precisely what Lemma~3 requires.
A third possibility is to change the ambient object. Instead of embedding (E, ∇)|U into a fixed Picard–Fuchs connection ℋ and then cutting it out by a projector, one could attempt to construct a Picard–Fuchs object to (E, ∇)|U (for instance via a linearization of non-abelian Hodge theory, or via a motivic Galois group attached to the Betti representation). Such a canonical ambient object might come equipped with a distinguished integral structure, making the summand projector unnecessary. At present, we do not know a general construction of this kind beyond the case where (E, ∇) is already Gauss–Manin.
Theorem~B gives a partial extension by enlarging the class of objects to the Tannakian subcategory generated by Picard–Fuchs objects requiring that the semisimple factors be cut out by motivated idempotents after inverting a single N. It is natural to ask whether the statement can be formulated intrinsically for semisimple objects of the Tannakian envelope without any mention of projectors, or whether one can at least treat semisimple without assuming they are direct summands of a Gauss–Manin system over R[1/N].
The obstruction is again integral control. A semisimple subquotient in MIC(U/K) need not admit an integral model over R[1/N] that is functorial in families, even when the ambient object does. Moreover, the passage from an extension of filtered connections to a filtration on a quotient requires strictness properties that are nontrivial in mixed situations. In the present paper we avoided these issues by restricting to semisimple direct summands defined by filtered idempotents, where strictness is automatic.
A promising intermediate goal is the following. Suppose we have an
exact sequence in MIC(U/R[1/N])
0 → (E1, ∇1) → (E, ∇) → (E2, ∇2) → 0
with (E, ∇) Picard–Fuchs (or
more generally satisfying Lam–Litt integrality) and with E1, E2
semisimple over K. If the
formal isomonodromic deformation of (E, ∇) is integral, then Lam–Litt
yields extension of filtrations on (E, ∇) along the formal leaf. One
may ask for a criterion ensuring that the induced filtrations on E1 and E2 also extend, possibly
after semisimplification. A purely formal argument would require
strictness of the filtration with respect to the morphisms in the exact
sequence; such strictness is familiar in the theory of admissible
variations of mixed Hodge structure, but here the filtration produced by
Lam–Litt is a nonlinear arithmetic avatar and is not known to satisfy
comparable exactness properties in general.
From the Tannakian viewpoint, one expects that semisimplicity should allow us to reduce to direct summands after passing to the semisimplification of the category. However, semisimplification does not commute with integral structures: the semisimplification of an integral object may require inverting additional primes, and the needed uniformity in N is precisely what our motivic hypotheses were designed to guarantee. Thus, any extension from summands to subquotients appears to demand new input: either a stronger integrality theorem asserting uniform semisimplicity modulo 𝔭 along the isomonodromy leaf, or a more flexible finiteness principle that does not require the full extension of filtrations for the object under consideration.
We emphasize that the end goal would be a statement of the following form: if (E, ∇)|U belongs to the semisimple Tannakian subcategory generated by Picard–Fuchs objects and its formal isomonodromic deformation is integral, then its isomonodromy leaf is algebraic. Achieving this would turn Theorem~A into a truly Tannakian nonlinear p-curvature principle, with the geometric content encoded entirely by membership in a well-behaved category, rather than by explicit correspondences.
Theorem~A concerns a single point of MdR(X/S, r) and the leaf of the isomonodromy foliation through that point. It is natural to ask whether similar techniques can be applied to subvarieties invariant under the foliation, or to the Zariski closure of the union of leaves satisfying a given arithmetic constraint.
One motivating analogy is the classical Noether–Lefschetz locus for variations of Hodge structure: loci where additional Hodge classes appear are algebraic, and in favorable situations have controlled components. In non-abelian Hodge theory, Simpson introduced non-abelian analogues (``non-abelian Hodge loci’’) inside moduli of local systems, defined by conditions such as admitting a Hodge filtration with specified Hodge numbers. Our argument uses a local version of such a condition: extension of a filtration along the formal isomonodromy deformation forces finiteness of the orbit. One may ask whether there is an algebraicity statement for the locus of points in MdR(X/S, r) whose formal isomonodromy leaf carries an integral filtered structure of the Lam–Litt type, possibly with fixed numerical invariants.
A difficulty is that the integral condition in Theorem~A is not obviously Zariski-constructible: it is formulated in terms of descent of a formal crystal to ŜR, s[1/N]. Nevertheless, one may hope for a ``spreading out’’ principle: if an algebraic family of de Rham points has integral formal isomonodromy at a Zariski-dense set of points of the base, then the family is contained in an algebraic union of special subvarieties (finite over the base). Any such statement would amount to a global version of the mechanism behind Lemma~4, and would provide a bridge from arithmetic integrality to the geometry of foliations on moduli.
Another direction is to study invariant subvarieties on the Betti side. The action of π1(S) on MB(Xs, r) is algebraic, and one may consider orbit closures, invariant subvarieties, and their relation to the de Rham foliation under Riemann–Hilbert. Theorem~A identifies algebraic leaves with finite orbits, hence with zero-dimensional orbit closures. It is natural to ask what arithmetic conditions force higher-dimensional orbit closures to be ``special’’ in some sense (for example, to be unions of translated subtori in the rank-one case, or to arise from representations with extra endomorphisms in higher rank). Any progress here would require moving beyond finiteness to a finer description of dynamical behavior under outer monodromy.
Finally, we explain how Theorem~A fits into a broader circle of conjectures inspired by Grothendieck–Katz. In the linear theory, vanishing (or nilpotence) of p-curvatures for almost all primes is expected to force finite monodromy, hence algebraicity of horizontal sections. In our nonlinear setting, the object of interest is not a single connection on a fixed base, but the isomonodromy foliation on a moduli space of connections. The natural analogue of ``vanishing p-curvature’’ is less clear, but the integrality of the formal isomonodromic deformation may be viewed as a first arithmetic constraint in this direction: it asserts that the formal leaf carries a compatible R[1/N]-model, hence admits meaningful reduction mod p for almost all p.
A speculative ``nonlinear p-curvature conjecture’’ would assert that sufficiently strong arithmetic constraints on the formal isomonodromy leaf force algebraicity of the leaf. One possible formulation is:
If a point of MdR(X/S, r) has
the property that its formal isomonodromic deformation extends over
ŜR, s[1/N]
and satisfies a suitable p-adic constraint for almost all
primes (for instance, admits a Frobenius structure, or yields
nilpotent/nonexpanding ``p-curvature’’ invariants along the
leaf), then the isomonodromy leaf is algebraic.
Theorem~A proves such a principle in a restricted ``geometric’’ range: we assume the point arises from a motivated Picard–Fuchs summand, and then integrality suffices because Lam–Litt provides the missing p-adic filtration-extension input. To move beyond this range, one would need a replacement for Lam–Litt applicable to broader classes of connections, or a different route from integrality to finiteness that avoids filtration theory altogether.
One attractive possibility is that, in the presence of semisimplicity, integrality of the formal leaf might imply that the induced p-adic representations (obtained by suitable companions, when they exist) are potentially unramified and bounded in a way that forces the orbit to be finite by a p-adic analytic argument. Another possibility is to exploit rigidity properties of the isomonodromy foliation itself: if the leaf admits good reductions mod p for infinitely many p and the reductions satisfy a Frobenius-periodicity condition, then the leaf might be forced to be algebraic by a form of p-adic algebraization. At present these ideas remain heuristic, but they suggest that the integrality hypothesis in Theorem~A may be the visible part of a stronger arithmetic structure.
We also note two directions where a full conjecture would have to go beyond the present paper. First, it should treat non-proper or logarithmic families systematically, where isomonodromy is most classically studied and where one expects braid/mapping class group dynamics to be richer. Second, it should address non-semisimple points, where orbit finiteness is not the correct conclusion and where one expects instead that arithmetic constraints force the representation to lie in a proper invariant subvariety (e.g. a unipotent or solvable locus). Extending our methods to these settings would require new input both from p-adic Hodge theory and from the geometry of moduli of connections with singularities.
In summary, Theorem~A should be viewed as evidence for a general philosophy: arithmetic integrality constraints on isomonodromic deformations impose strong algebro-geometric restrictions on the leaves of the isomonodromy foliation, and hence on the dynamics of monodromy actions on character varieties. The main technical challenge in further developments is to isolate an intrinsic, verifiable replacement for the motivated-projector hypothesis that still allows one to propagate arithmetic filtrations (or comparable structures) to the objects of interest.