The guiding theme of this work is that, even when the topological data of a flat bundle are held fixed, the underlying holomorphic bundle obtained by varying the complex structure should behave ``generically’’ from the Brill–Noether point of view. Concretely, we fix a rank-2 unitary monodromy representation on a punctured surface and let the complex structure on the surface vary: the associated logarithmic connection varies isomonodromically, hence its monodromy is constant, but its Deligne canonical extension produces a genuinely varying holomorphic bundle on the varying curve. Our main goal is to show that for a very general complex structure in the deformation space, this holomorphic bundle does not acquire unexpected low-degree sections.
This is a horizontal genericity statement: the deformation direction is constrained to lie in an isomonodromy leaf rather than being arbitrary in the moduli space of bundles. In particular, standard Brill–Noether results for bundles do not apply directly, because the holomorphic bundle in an isomonodromic family is highly non-linear in the parameter and retains strong analytic structure coming from the flat connection. Nevertheless, for unitary rank-2 systems of nontrivial irreducible type, the interaction between Kodaira–Spencer deformation of the curve and the (1, 0)-part of the connection is sufficiently transverse to force generic Brill–Noether vanishing.
We refer to the vanishing phenomenon we prove as . It asserts that
after twisting by a moving point p on the curve, the space of
holomorphic sections vanishes generically along the isomonodromic
deformation. More precisely, for very general parameter values and for a
general point p on the
corresponding fiber, one has
H0(Xt, Ẽt(p)) = 0,
where Ẽt
is the Deligne canonical extension of the isomonodromically varying
connection on Xt. This
formulation is well-adapted to semicontinuity and Brill–Noether type
arguments, since it isolates the lowest twist for which a section could
appear for a degree-0 rank-2 bundle.
A convenient equivalent form uses generic global generation. By Serre duality, a nonzero section of Ẽt(p) corresponds to a nontrivial quotient of Ẽt∨ ⊗ ωXt(Dt) supported at p, and thus the failure of global generation at p. Under mild cohomological hypotheses (satisfied in our setting for very general t), vanishing of H0(Ẽt(p)) for general p is equivalent to Ẽt∨ ⊗ ωXt(Dt) being generated by global sections at the generic point. We emphasize this second formulation because it is the precise input needed for Hodge-theoretic fixed-part arguments (as in the framework of infinitesimal variations of Hodge structure).
The isomonodromic viewpoint places our results in close conceptual proximity to mapping class group dynamics. Fixing a unitary representation ρ of π1 on the topological surface with punctures, varying the complex structure corresponds to moving in Teichm"uller space, and the mapping class group acts on the representation variety. When one descends from Teichm"uller space to the moduli stack of curves, the constancy of monodromy along an isomonodromic family reflects the fact that we remain in a single mapping class group orbit of ρ. From this perspective, our theorem can be read as a statement that, for an irreducible nontrivial unitary ρ in rank 2, the holomorphic bundles produced by the Deligne extension along the associated orbit are generically Brill–Noether general in the specific sense of having no sections after a degree-1 twist. Although we do not pursue dynamical consequences here, this perspective clarifies why one should expect ``genericity’’ even under the constraint of fixed monodromy: irreducibility and nontriviality prevent the orbit from being trapped in special Brill–Noether loci.
Our results are directly motivated by questions formulated in Litt’s
survey on families of varieties and the behavior of cohomology with
local system coefficients. In that setting, one seeks criteria forcing
the of a family of local systems to vanish, i.e. forcing
H0(M, R1π*𝕌) = 0
for a family of curves π : 𝒞 → M and a unitary
local system 𝕌 on 𝒞 \ 𝒟. Litt isolates a mechanism
(implemented, for instance, in ) whereby generic global generation of a
certain adjoint-type twist of the Deligne extension implies fixed-part
vanishing; see in particular the role played by and the surrounding
discussion. The conjectural picture (cf. ) predicts that for families of
curves sufficiently general in moduli, nontrivial unitary coefficient
systems should have no fixed part, reflecting the absence of
mapping-class-invariant cohomology in nontrivial local systems. Our main
technical contribution is to supply, in rank 2, a geometric input (horizontal generic
vanishing / generic global generation) that is designed precisely to
trigger Litt’s fixed-part vanishing mechanism.
We now state our results in the form that will be proved in the body of the paper. The first theorem treats the compact case D = ∅; it already exhibits the key analytic phenomenon without parabolic bookkeeping.
The logarithmic case is the one relevant for applications to pointed curves and to coefficient systems on 𝒞 \ 𝒟. Here the holomorphic bundle on X is not uniquely determined by the monodromy alone unless one specifies a canonical extension across the punctures. The Deligne canonical extension provides the needed canonical choice; in the unitary setting it is characterized by the condition that residue eigenvalues lie in the standard interval. With this normalization in place, the same generic vanishing holds with ωXt(Dt) replacing ωXt.
Theorem~ is the form we use to deduce fixed-part vanishing in families dominating moduli. The point is that in such a family the Kodaira–Spencer map is generically as large as possible, so the local analysis on a Kuranishi base can be transported to the given family after shrinking the base. The conclusion is that the invariants in the degree-1 cohomology with coefficients in 𝕌 vanish.
In the background, the proof strategy for Theorems~ and~ is a transversality argument against a Brill–Noether-type locus inside the parameter space. The condition H0(Xt, Ẽt(p)) ≠ 0 is closed by semicontinuity; thus, to obtain generic vanishing it suffices to show that it is not identically satisfied along the isomonodromic family. The key analytic input is an explicit formula for the first-order variation of the $\bar\partial$-operator defining the holomorphic structure on Ẽt in terms of a Beltrami differential μ representing the deformation of complex structure and the (1, 0)-part of the flat connection. This reduces the problem to constructing a Kodaira–Spencer class whose induced obstruction to deforming a putative section is nonzero. Irreducibility and nontriviality of the unitary representation are exactly what permit such ``killing directions’’ to exist: in reducible or trivial cases, sections persist for representation-theoretic reasons and the Brill–Noether locus may contain entire isomonodromic leaves.
We finally note two boundaries of the method. First, the rank-2 hypothesis is used in a substantive way to control the space of potential sections and to ensure that the relevant Brill–Noether condition behaves as expected; extensions to higher rank would require new ideas or stronger stability input. Second, the genus condition g ≥ 3 is not merely cosmetic: in low genus, special geometry of the moduli of curves can produce exceptional families with persistent sections, and the transversality argument can fail for reasons unrelated to monodromy. Within these constraints, the results above provide a concrete instance of Litt’s general philosophy: for families of curves varying maximally in moduli, unitary coefficient systems of nontrivial type do not contribute to the fixed part, and this vanishing is ultimately controlled by generic global generation properties of the associated Deligne extensions along isomonodromic deformations.
We collect the standard correspondence results and extension procedures that underlie our later deformation-theoretic arguments. Throughout, X is a smooth complex projective curve, $D=\sum_{i=1}^n x_i$ is a reduced effective divisor (possibly empty), and we write X∘ := X \ D.
A rank-2 unitary local system on
X∘ may be described
equivalently as a representation
ρ : π1(X∘) → U(2)
up to conjugacy, or as a C∞ complex vector bundle
V → X∘ of
rank 2 equipped with a flat unitary
connection ∇ (i.e. with holonomy
contained in U(2)). Fixing a
C∞ identification
of the underlying surface, the representation ρ determines (V, ∇) uniquely up to unitary
gauge.
Given a complex structure on the underlying surface, a unitary
connection canonically splits as
∇ = ∇1, 0 + ∇0, 1.
We obtain a holomorphic bundle structure on the underlying C∞ bundle by
setting
$$
\bar\partial_E:=\nabla^{0,1},
$$
since flatness implies (∇0, 1)2 = 0. Thus, for
each complex structure on the surface (and for each choice of
punctures), the topological local system yields a generally holomorphic
bundle. This is the basic mechanism behind all ``horizontal’’
Brill–Noether phenomena: monodromy may be held fixed, while the
holomorphic structure varies because ∇0, 1 depends on the complex
structure.
In rank 2, the irreducibility of ρ implies that the associated holomorphic bundle is stable (or, in the punctured case, parabolically stable) once the extension data at D are fixed. Nontriviality excludes the split case V ≅ ℂ2 with trivial holonomy, where holomorphic sections persist for elementary reasons.
To work algebro-geometrically on X rather than on X∘, we need a canonical
extension across D. For
unitary ρ, local monodromy
around each puncture is semisimple with eigenvalues on the unit circle;
in particular, the corresponding flat bundle has regular singularities.
Concretely, in a holomorphic coordinate z centered at xi and on a
suitable local trivialization over a punctured disk, one may write the
connection as
$$
\nabla = d + A\,\frac{dz}{z} + (\text{holomorphic terms}),
$$
where A is a constant matrix
whose eigenvalues are logarithms of the monodromy eigenvalues. Since
monodromy is unitary, these eigenvalues have purely imaginary parts
modulo ℤ, and one may choose a
representative with real parts in [0, 1). This choice is precisely what Deligne
uses to single out a distinguished extension.
Let (V, ∇) be a flat bundle
on X∘ with regular
singularities along D. The is
the unique logarithmic connection (E, ∇) on X extending (V, ∇) such that, at each xi, all
eigenvalues of the residue
Resxi(∇) ∈ End (E|xi)
have real parts in the interval [0, 1).
In the unitary case, existence and uniqueness are standard and can be
seen explicitly in local models: if ∇ = d + A dz/z
with A diagonalizable and
ℜ(spec(A)) ⊂ [0, 1), then the
𝒪-lattice generated by multivalued flat
sections multiplied by z−A is
single-valued and yields the canonical extension. We emphasize that,
although the flat bundle is determined by ρ, the extension across D is not a formal consequence of
monodromy unless one imposes such a normalization; our later
cohomological statements concern the holomorphic bundle underlying this
specific extension.
When D = ∅, the Deligne extension is vacuous and one simply has a holomorphic bundle with a flat unitary connection on X. When D ≠ ∅, we will always tacitly regard (E, ∇) as the Deligne extension; in particular, residues are fixed with eigenvalues in [0, 1) and do not jump under small deformations as long as monodromy conjugacy classes remain fixed.
In the compact case D = ∅, the Narasimhan–Seshadri theorem identifies irreducible unitary representations with stable vector bundles of degree 0. More precisely, an irreducible representation ρ : π1(X) → U(2) determines a stable holomorphic bundle E of degree 0 together with a projectively flat unitary connection, and conversely every stable degree-0 bundle arises in this way from a unitary representation. For our purposes, the key consequence is that irreducibility of ρ implies stability of the corresponding holomorphic bundle for each fixed complex structure on X.
In the pointed case, the correct replacement is the Mehta–Seshadri
correspondence. Fix conjugacy classes for local monodromy around each
xi. A
unitary representation ρ : π1(X∘) → U(2)
with prescribed local conjugacy classes corresponds to a polystable
bundle E* on (X, D) of parabolic degree
0, with parabolic weights determined by
the monodromy eigenvalues (equivalently, by the residue eigenvalues in
the Deligne interval). In rank 2, the
parabolic structure at xi consists of a
flag
E|xi = Fi, 1 ⊋ Fi, 2 ⊋ 0
together with weights 0 ≤ αi, 1 ≤ αi, 2 < 1,
where αi, j
are the residue eigenvalues of the Deligne extension (counted with
multiplicity). If the local monodromy has distinct eigenvalues, the flag
is uniquely determined by the eigenspace decomposition of the residue;
if the local monodromy is scalar, the parabolic data become less rigid,
but in the unitary Deligne-normalized situation the underlying extension
is still canonical.
The passage from (E, ∇) to a parabolic bundle may be described concretely: the residue endomorphism at each xi induces a filtration of the fiber by generalized eigenspaces, and the corresponding eigenvalues in [0, 1) serve as weights. Conversely, given a stable parabolic bundle of parabolic degree 0, pardeg(E*) = 0, Mehta–Seshadri produces a unitary representation of π1(X∘) with the specified local monodromy. For us, the important point is that irreducibility of ρ implies parabolic stability of E*, and hence good deformation behavior: parabolic endomorphisms are scalars, and Brill–Noether loci cut out by parabolic section conditions behave as expected (in particular, they do not coincide with entire deformation spaces unless forced by representation-theoretic trivialities).
We next recall the notion of isomonodromic deformation tailored to
our later use. Let (𝒳, 𝒟) → T
be a deformation of the pointed curve (X, D) over a small
analytic base T (typically a
Kuranishi space). After shrinking, we may and will assume T is contractible and that π : 𝒳 → T is a C∞ locally trivial
fibration, with 𝒟 ⊂ 𝒳 a disjoint union
of sections. In particular, the complements
𝒳∘ := 𝒳 \ 𝒟, Xt∘ := Xt \ Dt
form a C∞ fiber
bundle over T, and choosing a
basepoint identifies π1(Xt∘)
with π1(X∘)
for all t.
Fix a unitary representation ρ : π1(X∘) → U(2). Over 𝒳∘ we obtain a rank-2 unitary local system $\widetilde{\mathbb U}$ by transporting ρ along the above identifications; equivalently, $\widetilde{\mathbb U}$ is the pullback of the constant local system on T with fiber the representation space, equipped with the π1-action prescribed by ρ on each fiber. By construction, the restriction $\widetilde{\mathbb U}|_{X_t^\circ}$ has monodromy ρ for every t.
Passing to vector bundles with connection, $\widetilde{\mathbb U}$ corresponds to a
C∞ bundle Ṽ → 𝒳∘ with a flat
unitary connection $\widetilde\nabla$
(flat along directions on 𝒳∘, not merely relatively over
T). One then seeks an
extension across 𝒟 compatible with the
complex structures on fibers. The relevant object is a logarithmic
connection on 𝒳 with poles along 𝒟,
$$
(\widetilde E,\widetilde\nabla):\qquad \widetilde\nabla:\widetilde
E\longrightarrow \widetilde E\otimes \Omega^1_{\mathcal X}\bigl(\log
\mathcal D\bigr),
$$
whose restriction to 𝒳∘ is
the flat connection on Ṽ and
whose restriction to each fiber (Xt, Dt)
is the Deligne canonical extension of the induced local system on Xt∘.
Existence and uniqueness of such $(\widetilde E,\widetilde\nabla)$ over a sufficiently small T are standard consequences of the logarithmic Riemann–Hilbert correspondence in families (or, more concretely, by extending locally near each section of 𝒟 using the Deligne lattice determined by the fixed local monodromy). Because T is contractible and the conjugacy classes of local monodromy are constant, the residue eigenvalues remain in the same Deligne interval [0, 1) and therefore the canonical extension varies holomorphically with t. We will refer to $(\widetilde E,\widetilde\nabla)$ as the of (E, ∇) over T.
Two features of this construction are crucial for what follows. First, the monodromy representation on each Xt∘ is fixed by construction, so all variation occurs at the level of holomorphic structure on the Deligne extension bundles Ẽt. Second, the dependence of Ẽt on t is far from algebraic in general: it is governed by the interaction between the (1, 0)-part of the flat connection and the deformation of complex structure. This is precisely what allows one to formulate and prove transversality statements against Brill–Noether type conditions along the isomonodromy leaf.
In the next section we set up the deformation framework in which we work: we choose a local universal deformation (𝒳, 𝒟) → T, construct $(\widetilde E,\widetilde\nabla)$ as above, and reduce the main vanishing statement to a local analytic problem on T involving the first-order variation of the induced $\bar\partial$-operators and the geometry of the corresponding Brill–Noether loci in T × X.
We now fix the deformation-theoretic setting in which the horizontal vanishing statement will be proved. All arguments are local on the base, and we work analytically throughout.
Let (X, D) be as in the enclosing scope, with $D=\sum_{i=1}^n x_i$ reduced (possibly n = 0). LetWe write (Xt, Dt) for the fiber over t ∈ T, and Xt∘ := Xt \ Dt for the punctured fiber.
By standard deformation theory, the Kodaira–Spencer map identifies
the tangent space at the origin with the log-tangent cohomology:
T0T ≅ H1(X, TX(−D)),
and universality means that every small deformation of (X, D) is pulled back from
(𝒳, 𝒟) → T uniquely up to
isomorphism. We will repeatedly use the fact that we may test
``generic’’ behavior on T by
producing a single tangent direction κ ∈ H1(X, TX(−D))
with the required transversality property at 0.
In addition, by Ehresmann’s theorem, after shrinking T there exists a C∞ locally trivialization
of the family. Concretely, we may fix a C∞ manifold Σ (the underlying oriented surface
of X), a finite subset P ⊂ Σ corresponding to
D, and C∞ diffeomorphisms
$$
\Phi_t:(\Sigma,P)\stackrel{\sim}{\longrightarrow}(X_t,D_t)
$$
varying smoothly in t, with
Φ0 the identity
under our fixed identification of (X, D) with the central
fiber. Since T is
contractible, this provides canonical identifications
π1(Xt∘) ≅ π1(X∘)
up to inner automorphism; for our purposes (unitary representations up
to conjugacy) this ambiguity is harmless and will be suppressed.
Let ρ : π1(X∘) → U(2)
be irreducible and nontrivial, and let (E, ∇) denote the associated Deligne
canonical logarithmic flat bundle on (X, D) as in . Over 𝒳∘ := 𝒳 \ 𝒟 we first construct the
isomonodromic object: using the identifications of fundamental groups
along T, we obtain a
rank-2 local system
$$
\widetilde{\mathbb U}\;\text{ on }\;\mathcal X^\circ
$$
whose restriction to each fiber Xt∘
has monodromy ρ. Equivalently,
we obtain a C∞
complex vector bundle Ṽ → 𝒳∘ with a flat
unitary connection $\widetilde\nabla$
such that $(\widetilde
V,\widetilde\nabla)|_{X_t^\circ}$ corresponds to ρ for all t.
The essential point is to extend $(\widetilde V,\widetilde\nabla)$ across
𝒟 in a way compatible with the complex
structures on the fibers and with Deligne’s normalization of residues.
For each component 𝒟i we may choose, locally
on 𝒳 near 𝒟i, a holomorphic
coordinate zi along the
fibers with 𝒟i = {zi = 0};
by regular singularity and unitarity, the connection has a local normal
form
$$
\widetilde\nabla \;=\; d + A_i\,\frac{dz_i}{z_i} + (\text{holomorphic
}1\text{-form terms}),
$$
with Ai
semisimple and ℜ(spec(Ai)) ⊂ [0, 1)
(Deligne interval). Because the monodromy conjugacy class around 𝒟i is constant in t by construction, we may take Ai to be
constant along T (after
possibly changing trivialization), and the eigenvalues in [0, 1) do not jump.
We then define the Deligne lattice near 𝒟i by the usual
prescription: if s is a
multivalued flat frame on the punctured neighborhood, we set
e := zi−Ais,
which is single-valued precisely because Ai records the
logarithm of the fixed unitary local monodromy with eigenvalues chosen
in [0, 1). The 𝒪-module generated by the components of e yields a locally free extension
across 𝒟i, and the
connection extends as a logarithmic connection with residue Ai. Performing
this construction simultaneously near each 𝒟i and gluing with the
flat bundle on 𝒳∘ produces a
holomorphic vector bundle Ẽ on
𝒳 equipped with a logarithmic
connection
$$
\widetilde\nabla:\widetilde E\longrightarrow \widetilde E\otimes
\Omega^1_{\mathcal X}\bigl(\log \mathcal D\bigr)
$$
restricting to (E, ∇) on the
central fiber. Uniqueness holds in the expected sense: any two such
extensions differ by an automorphism of the flat bundle over 𝒳∘ preserving the Deligne lattices
at 𝒟, hence by a holomorphic gauge
transformation that is fiberwise flat; by irreducibility this
automorphism is scalar on fibers, so after fixing the identification
over t = 0 we obtain a
canonically defined isomonodromic deformation over T.
We emphasize what will matter later: the of $\widetilde\nabla|_{X_t^\circ}$ is constant, but the on the bundle Ẽt := Ẽ|Xt varies nontrivially with t. Indeed, Ẽt is the Deligne extension of a fixed C∞ unitary local system with respect to the varying complex structure on the fiber.
Our vanishing statement involves Ẽt(p), where p ∈ Xt varies. It is useful to package this into a family over the total space 𝒳.
Let 𝒳×T𝒳 be the
fiber product, with projections p1, p2
and diagonal Δ ⊂ 𝒳×T𝒳. The
line bundle 𝒪(Δ) is relatively
Cartier over T, and for each
t its restriction to Xt × Xt
corresponds to the usual divisor of the diagonal. We consider the vector
bundle on 𝒳×T𝒳
ℱ := p1*Ẽ ⊗ 𝒪(Δ).
For a point p ∈ Xt,
restricting ℱ to Xt × {p}
identifies canonically with Ẽt(p),
i.e. with Ẽt ⊗ 𝒪Xt(p).
Thus questions about H0(Xt, Ẽt(p))
become questions about fiberwise H0 of ℱ under the projection p2 : 𝒳×T𝒳 → 𝒳.
This formulation has two advantages. First, it makes the dependence on the point p holomorphic. Second, it allows us to use semicontinuity and base-change for the coherent sheaf (p2)*ℱ.
Define the subset
𝒲 := {(t, p) ∈ 𝒳 | H0(Xt, Ẽt(p)) ≠ 0}.
Equivalently, using the preceding discussion,
(t, p) ∈ 𝒲 ⇔ H0(Xt × {p}, ℱ|Xt × {p}) ≠ 0.
By Grauert’s semicontinuity theorem applied to the proper holomorphic
map p2 : 𝒳×T𝒳 → 𝒳
and the coherent sheaf ℱ, the
function
(t, p) ↦ h0(Xt, Ẽt(p))
is upper semicontinuous on 𝒳 in the
analytic topology. In particular, 𝒲 is
a closed analytic subset of 𝒳.
Our main vanishing statement over T is equivalent to the claim that 𝒲 is a proper subset of 𝒳 and does not dominate T: for very general t, the fiber 𝒲 ∩ Xt is a proper closed subset of Xt, hence a finite union of points, and for general p ∈ Xt we have H0(Xt, Ẽt(p)) = 0.
Accordingly, the horizontal generic vanishing theorem will follow once we show that 𝒲 ≠ 𝒳. Since 𝒲 is closed analytic, it suffices to exhibit a single pair (t, p) with H0(Xt, Ẽt(p)) = 0. However, we will not attempt to find such a pair by an explicit global construction. Instead, we argue by deformation: assuming that (0, p0) ∈ 𝒲 for some point p0 ∈ X, we study the persistence of a section under infinitesimal variation of complex structure.
More precisely, fix (0, p0) ∈ 𝒳 and suppose s ∈ H0(X, E(p0)) is a nonzero section. If 𝒲 = 𝒳, then for every tangent vector κ ∈ T0T the section s admits a first-order extension along the corresponding first-order deformation of (X, D), after possibly moving p0 in the fiber direction. Conversely, if we can find Kodaira–Spencer class κ such that s is obstructed to first order along κ (in a sense made precise later), then 𝒲 cannot contain a full neighborhood of (0, p0), and therefore cannot coincide with 𝒳.
Thus the proof reduces to an infinitesimal transversality problem: given a putative section of E(p0), we seek a deformation direction of the pointed curve which kills it. The key input is that our deformation is : the underlying C∞ unitary connection is fixed, and only the induced $\bar\partial$-operator changes with the complex structure. Consequently, the first-order obstruction to deforming s may be computed explicitly from the first variation of $\bar\partial$ in terms of κ and the (1, 0)-part of the flat connection. Establishing such a formula is the purpose of the next section.
In the isomonodromic situation the underlying C∞ data are constant: we have a fixed rank-2 complex vector bundle V on the underlying oriented surface Σ (with marked set P) equipped with a flat unitary connection ∇, and for each t ∈ T we transport this pair to Xt∘ using the C∞ trivialization of the family. The holomorphic bundle Ẽt is then obtained by declaring that the (0, 1)–part of ∇ with respect to the complex structure of Xt is the Dolbeault operator. The purpose of this section is to make this dependence on t explicit and to record a coordinate-level first-variation formula in terms of Beltrami differentials.
Fix t ∈ T. Let
Jt denote
the complex structure on Σ
corresponding to the fiber Xt, and
write
$$
A^1(\Sigma,\End V)\;=\;A^{1,0}_{J_t}(\End V)\;\oplus\;A^{0,1}_{J_t}(\End
V)
$$
for the type decomposition of 1-forms
with values in $\End(V)$ with respect
to Jt (and
similarly for V-valued forms).
Decompose the connection
∇ = ∇t1, 0 + ∇t0, 1,
where ∇t0, 1 : A0(Σ, V) → AJt0, 1(Σ, V)
is C-linear. Since ∇ is flat, we have ∇2 = 0, hence (∇t0, 1)2 = 0,
and therefore
$$
\bar\partial_t \;:=\;\nabla^{0,1}_{t}
$$
defines an integrable holomorphic structure on V over Xt∘.
The resulting holomorphic bundle is precisely the restriction of Ẽt to Xt∘
(and in the logarithmic case, it is the holomorphic structure on the
Deligne lattice away from 𝒟t).
Two remarks will be useful later. First, although ∇ is fixed, the splitting into types depends on Jt, so the operator $\bar\partial_t$ varies nontrivially with t. Second, by construction ∇ is a holomorphic connection for $\bar\partial_t$ in the sense that $\nabla^{0,1}_{t}=\bar\partial_t$, i.e. ∇ is of type (1, 0) + (0, 1) with the prescribed (0, 1)–part.
We recall the standard description of the tangent space to the
deformation space in analytic terms. A first-order deformation of the
complex structure of X
preserving the marked divisor D is represented by a Beltrami
differential
μ ∈ A0, 1(X, TX(−D)),
well-defined up to $\bar\partial$-exact
terms. Under the Kodaira–Spencer identification T0T ≃ H1(X, TX(−D)),
the class [μ] corresponds to
the tangent vector κ.
Concretely, on a holomorphic coordinate chart (U, z) for X (centered at a point of U), the tensor μ can be written as
$$
\mu\;=\;\mu_{\bar z}^{z}\,d\bar z\otimes \frac{\partial}{\partial z},
$$
where μz̄z
is a smooth function on U; if
U meets D and z is chosen with D ∩ U = {z = 0},
then the condition μ ∈ A0, 1(TX(−D))
is equivalent to the vanishing μz̄z(0) = 0
(i.e. the vector field component has a factor of z), reflecting the fact that the
marked point is preserved in the pointed deformation.
Given a small parameter ε
with ε2 = 0, the
deformed complex structure Jε determined by
μ is characterized locally by
the fact that the new (0, 1) tangent
direction is spanned by
and correspondingly the (0, 1)
cotangent direction is spanned by a 1-form of the shape
up to terms of order ε2 (which vanish). We
will only use the first-order identities –.
We now compute $\bar\partial_\varepsilon$ in terms of $\bar\partial_0$ and μ. Work on a coordinate chart (U, z) on X and choose a C∞ trivialization of
V over U so that the connection takes the
form
$$
\nabla \;=\; d + \Gamma,
\qquad \Gamma\in A^1(U,\End V).
$$
Write Γ = Γz dz + Γz̄ dz̄.
For a C∞ section
s of V on U,
$$
\nabla s \;=\; \left(\frac{\partial s}{\partial z}+\Gamma_z s\right)dz
\;+\;
\left(\frac{\partial s}{\partial \bar z}+\Gamma_{\bar z} s\right)d\bar
z.
$$
With respect to the complex structure J0 (i.e. at ε = 0) we have
$$
\bar\partial_0 s \;=\; \left(\frac{\partial s}{\partial \bar
z}+\Gamma_{\bar z}s\right)d\bar z,
\qquad
\nabla^{1,0}_0 s \;=\; \left(\frac{\partial s}{\partial z}+\Gamma_z
s\right)dz.
$$
For the deformed complex structure Jε, the operator
$\bar\partial_\varepsilon$ is obtained
by projecting ∇s to type (0, 1) with respect to Jε.
Equivalently, we may evaluate ∇s on the deformed (0, 1) vector field and then attach the
deformed (0, 1) form. To first
order,
The first term in is $\bar\partial_0
s$. For the second term we recognize the contraction of μ with the (1, 0)–part of ∇:
$$
\mu \,\lrcorner\, \nabla^{1,0}_0 s
\;=\;
\mu_{\bar z}^{z}\,d\bar z\otimes \frac{\partial}{\partial
z}\,\lrcorner\,
\left(\nabla_{\partial/\partial z}s\right)dz
\;=\;
\mu_{\bar z}^{z}\left(\nabla_{\partial/\partial z}s\right)d\bar z.
$$
Thus we obtain the first-order formula
in the sense that for every C∞ section s we have
$$
\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}\bar\partial_\varepsilon(s)
\;=\;
\mu\,\lrcorner\,\nabla^{1,0}_0(s).
$$
Since the right-hand side is manifestly C-linear in s and tensorial in μ, this expression glues to give a
global operator
$$
\delta_\mu \bar\partial \;:\; A^0(X,V)\longrightarrow A^{0,1}(X,V),
\qquad
\delta_\mu \bar\partial(s)=\mu\,\lrcorner\,\nabla^{1,0}(s),
$$
depending C-linearly on μ.
It is convenient to interpret without coordinates. Regard μ as a (0, 1)-form with values in TX(−D). The operator ∇1, 0 is a (1, 0)-differential operator on sections of V with values in V ⊗ ΩX1(D) in the logarithmic setting (and in V ⊗ ΩX1 when D = ∅). Contracting the vector-field component of μ with the 1-form component of ∇1, 0s yields a (0, 1)-form with values in V, which is precisely $\delta_\mu \bar\partial(s)$.
Moreover, if we change μ by a $\bar\partial$-exact term, the induced first-order deformation of the complex structure is isomorphic, and correspondingly the induced deformation of the holomorphic structure on V is gauge-equivalent. At the level of the variation operator, one checks (by a standard computation using flatness of ∇) that if $\mu=\bar\partial \xi$ for a smooth vector field ξ vanishing along D, then μ ⌟ ∇1, 0 differs from the commutator $[\bar\partial,\nabla_\xi]$ by $\bar\partial$-exact terms; in particular, the cohomology class of μ ⌟ ∇1, 0s in H1(X, V) depends only on [μ] ∈ H1(X, TX(−D)). This is the form in which the obstruction class will appear later.
Finally, we explain briefly why the same formula applies to the
holomorphic bundles Ẽt on the fibers
Xt, not
merely on Xt∘.
Fix a marked point xi ∈ D
and choose a holomorphic coordinate z on X with xi = {z = 0}.
In a local trivialization adapted to the Deligne lattice, the
logarithmic connection has the form
$$
\nabla \;=\; d \;+\; A\,\frac{dz}{z} \;+\; \alpha,
$$
where A is constant semisimple
with $\Re(\spec A)\subset[0,1)$ and
α is a smooth $\End(V)$-valued 1-form with at worst logarithmic growth
compatible with regular singularity. The holomorphic structure on the
Deligne extension is still given by the (0, 1)–part of ∇ with respect to the complex structure; the
point is that $\bar\partial_t$ remains
regular at z = 0 because the
singular term A dz/z is
of type (1, 0) for every complex
structure and hence contributes only to ∇t1, 0.
For the variation formula, note that a Beltrami differential for
pointed deformations satisfies μz̄z(0) = 0.
Therefore the contraction
$$
\mu\,\lrcorner\,\left(A\,\frac{dz}{z}\right)
\;=\;
\mu_{\bar z}^{z}\,A\,\frac{d\bar z}{z}
$$
is in fact at z = 0 (since
μz̄z
vanishes to first order in z).
Consequently, the operator μ ⌟ ∇1, 0 maps smooth
sections of the Deligne lattice to smooth (0, 1)-forms with values in the same lattice,
and computes the first-order variation of the holomorphic structure on
Ẽt on the
entire compact curve.
We will use as the basic computational input for transversality: given a section of E(p), the class of μ ⌟ ∇1, 0(s) in H1(X, E(p)) measures the first-order obstruction to deforming s along the Kodaira–Spencer direction [μ].
We now isolate the Brill–Noether-type conditions that will play the role of ``bad loci’’ for the isomonodromic deformation. Throughout, we keep the standing hypotheses: the rank-2 monodromy is unitary, irreducible, and nontrivial, and (E, ∇) is the Deligne canonical extension in the logarithmic case.
For each fiber Xt the
holomorphic bundle Ẽt has degree
0 (and in the logarithmic case it
carries the parabolic structure determined by the residues in the
Deligne interval). By the Narasimhan–Seshadri (resp. Mehta–Seshadri)
correspondence, irreducibility of the unitary representation implies
that Ẽt is
stable as a bundle (resp. parabolic stable of parabolic degree 0). In particular,
for every t, since any nonzero
section would produce a degree ≥ 0
line subbundle, contradicting stability of a degree-0 rank-2
bundle. The equality is a key simplification: all Brill–Noether
phenomena we will consider arise only after twisting by a point.
Let π : 𝒳 → T be
our local universal family, and denote by Ẽ the holomorphic vector bundle on
𝒳 whose restriction to Xt is Ẽt. Consider the
subset
ℬ := {(t, p) ∈ 𝒳 | H0(Xt, Ẽt(p)) ≠ 0}.
We think of ℬ as a Brill–Noether locus
in the total space 𝒳, parametrizing
those points of fibers at which a section with at most a simple pole can
exist.
Two basic properties are immediate.
By upper semicontinuity of cohomology in proper holomorphic families,
(t, p) ↦ h0(Xt, Ẽt(p))
is upper semicontinuous on 𝒳. Hence
ℬ is a closed analytic subset of 𝒳.
Fix (t, p) ∈ 𝒳. The
short exact sequence on Xt
induces a connecting homomorphism
δt, p: Ẽt(p)|p → H1(Xt, Ẽt).
Using , the long exact sequence in cohomology shows
Thus (t, p) ∈ ℬ if
and only if δt, p
is injective.
To globalize δt, p,
we use the universal ``add a point’’ exact sequence. Let p1, p2 : 𝒳×T𝒳 → 𝒳
be the projections and Δ ⊂ 𝒳×T𝒳 the
relative diagonal. On 𝒳×T𝒳 we have
and p2*Ẽ(Δ)|Δ ≃ Ẽ ⊗ 𝒪Δ
canonically. Pushing forward along p1 yields a morphism of
vector bundles on 𝒳
whose fiber at (t, p)
identifies with the connecting map δt, p
(up to the standard identification Ẽt(p)|p ≃ Ẽt|p ⊗ TXt, p,
which is harmless for rank considerations). In particular, shows that
ℬ is the rank-dropping locus of the
bundle map . Consequently, ℬ is a
determinantal analytic subset of 𝒳.
Since all objects involved come from algebraic data (after restricting to a sufficiently small algebraic neighborhood of the basepoint in moduli), the same argument also shows that ℬ is algebraic in the appropriate sense: it is cut out by minors of a morphism of algebraic vector bundles on 𝒳.
Although we will only need the properness of ℬ, it is useful to keep in mind the expected
size of this locus. The vector bundles in have ranks
$$
\rk(\widetilde E)=2,\qquad \rk\bigl(R^1\pi_*\widetilde
E\bigr)=h^1(X_t,\widetilde E_t).
$$
By and Riemann–Roch, χ(Ẽt) = 2(1 − g),
so h1(Xt, Ẽt) = 2g − 2
for all t. Thus δ is a morphism from rank 2 to rank 2g − 2, and ℬ is the locus where δ has rank ≤ 1. For a such map, the expected
codimension is
(2 − 1)((2g − 2) − 1) = 2g − 3.
Since dim (𝒳) = dim (T) + 1 = (3g − 3 + n) + 1 = 3g − 2 + n,
this heuristic predicts dim (ℬ) ≈ g + 1 + n, in
particular ℬ should be a proper subset
as soon as g ≥ 3. We emphasize
that we are not asserting genericity of δ at this stage; the point is rather
that there is no reason for ℬ to
coincide with all of 𝒳.
It is also useful to reinterpret H0(Xt, Ẽt(p)) ≠ 0 in terms of subbundles. A nonzero section of Ẽt(p) yields a nonzero map 𝒪Xt → Ẽt(p), and after saturation we obtain a line subbundle L ⊂ Ẽt(p) with deg (L) ≥ 0. Twisting back, this is equivalently a line subbundle L(−p) ⊂ Ẽt of degree ≥ −1. Stability of Ẽt forces deg (L(−p)) ≤ −1, hence necessarily deg (L(−p)) = −1. Thus the Brill–Noether condition forces Ẽt to admit a maximal line subbundle of degree −1, and one may view ℬ as measuring the presence of such borderline subbundles.
Set
Ft := Ẽt∨ ⊗ ωXt(Dt), F := Ẽ∨ ⊗ ω𝒳/T(𝒟),
where ω𝒳/T(𝒟) denotes
the relative log-canonical bundle. We consider the failure of generic
global generation of Ft. Concretely,
define
𝒢 := {t ∈ T | Ft
is not generated by global sections at the generic point of
Xt}.
Equivalently, t ∈ 𝒢 if and
only if the evaluation map
$$
\ev_t:\ H^0(X_t,F_t)\otimes \mathcal O_{X_t} \longrightarrow F_t
$$
fails to be surjective on every nonempty Zariski open subset of Xt.
The basic geometric object controlling this is again a determinantal
locus. Over 𝒳 there is a natural
evaluation morphism
$$
\ev:\ \pi^*\bigl(\pi_*F\bigr)\longrightarrow F,
$$
defined wherever π*F is locally
free (after possibly shrinking T so that cohomology commutes with
base change). The subset of 𝒳 where
$\ev$ drops rank is closed analytic;
its image in T is precisely
𝒢. In particular, 𝒢 is a closed analytic subset of T.
For our later application we need to pass between the point-twist
Brill–Noether condition and generic generation. There is a standard
equivalence in this rank-2 situation
(recorded separately as Lemma~5 in our list of lemmas): under the
vanishing H1(Xt, Ft) = 0,
generic global generation of Ft is equivalent
to the vanishing H0(Xt, Ẽt(p)) = 0
for a general point p ∈ Xt.
In the present unitary stable setting, the hypothesis H1(Xt, Ft) = 0
holds uniformly in t: by Serre
duality,
H1(Xt, Ft)∨ ≃ H0(Xt, Ẽt(−Dt)),
and stability of Ẽt together with
deg (Ẽt(−Dt)) < 0
gives H0(Xt, Ẽt(−Dt)) = 0.
Thus the ``bad’’ locus for generic generation may be detected by the
existence of point-twist sections, i.e. by ℬ.
When D ≠ ∅, the natural stability statement is parabolic: Ẽt carries the parabolic filtration and weights determined by the residue eigenvalues of the Deligne canonical extension. From the algebro-geometric point of view, one may model this either in the category of parabolic bundles on (Xt, Dt) or, equivalently, as an honest vector bundle on the corresponding root stack. In either interpretation, the constructions above remain determinantal.
More explicitly, twisting by a point p ∈ Xt \ Dt
does not interact with the parabolic structure, so the locus
ℬ = {(t, p) ∣ H0(Xt, Ẽt(p)) ≠ 0}
is defined exactly as before and is again closed analytic by
semicontinuity. Likewise, the sheaf Ft = Ẽt∨ ⊗ ωXt(Dt)
is the correct one for Serre duality in the logarithmic setting and for
the subsequent Hodge-theoretic applications. The evaluation map for
Ft and its
rank-dropping locus are defined intrinsically on the compact curve Xt, and the
resulting non-generation locus is again a closed analytic (indeed
algebraic) subset.
The only additional subtlety in the parabolic context is that, if one
chooses to reformulate H0(Ẽt(p)) ≠ 0
in terms of subobjects, one should interpret
line subbundle'' asparabolic line subbundle’’ (or as a line
subbundle on the root stack). The Deligne interval condition $\Re(\spec \Res_{x_i}\nabla)\subset[0,1)$
guarantees that the parabolic weights lie in the standard range, so
degrees and stability inequalities behave in the usual way. For our
purposes, it suffices to note that the Brill–Noether loci remain proper
candidates for determinantal conditions against which we will test the
isomonodromic directions in the next section.
We now explain the key point needed to conclude that the Brill–Noether locus ℬ ⊂ 𝒳 is all of 𝒳. Since ℬ is already known to be a closed determinantal analytic subset, it suffices to show that at some point of ℬ the horizontal (isomonodromic) directions are not everywhere tangent to ℬ. Equivalently, we will produce an infinitesimal deformation of the pointed curve that obstructs the persistence of any would-be section of Ẽt(p).
To make the deformation-theoretic computation precise, we temporarily
rigidify the point p. Fix a
point (t0, p0) ∈ 𝒳
with p0 ∉ Dt0,
and consider the local deformation space of the pointed curve (Xt0, Dt0 + p0).
Its tangent space identifies with
T(t0, p0)(𝒳) ≃ H1(Xt0, TXt0(−Dt0 − p0)) ⊕ TXt0, p0,
where the first summand corresponds to deformations of complex structure
fixing the marked points Dt0
and the extra marked point p0, and the second
summand corresponds to moving p0 in the fixed curve.
Since our goal is to detect variation in the base direction, we will
work in the subspace H1(TXt0(−Dt0 − p0)).
Assume (t0, p0) ∈ ℬ, so H0(Xt0, Ẽt0(p0)) ≠ 0, and let s ∈ H0(Xt0, Ẽt0(p0)) be a nonzero section. We ask whether s can persist under a first-order deformation of (Xt0, Dt0 + p0) of the connection. Concretely, we fix the underlying C∞ bundle with its unitary flat connection, and we vary only the complex structure; the holomorphic structure on Ẽt0 is the (0, 1)-part of the fixed connection with respect to the varying complex structure.
Let κ ∈ H1(TXt0(−Dt0 − p0))
be a Kodaira–Spencer class, and choose a Beltrami representative μ ∈ A0, 1(TXt0(−Dt0 − p0)).
By Lemma~1 (first variation of $\bar\partial$ under fixed unitary
connection), the induced first-order variation of the holomorphic
structure on Ẽt0
is given by
$$
\delta_{\mu}\bar\partial_{\widetilde E}(u)\ =\ \mu\ \lrcorner\
\widetilde\nabla^{1,0}(u),
$$
for smooth sections u of the
underlying C∞
bundle (interpreting the contraction in the standard way). If we attempt
to deform s to a first-order
family sε = s + ε ṡ
with ε2 = 0, the
holomorphicity equation $\bar\partial_{\varepsilon}(s_\varepsilon)=0$
yields, at first order,
Thus the obstruction to solving is precisely the Dolbeault cohomology
class
which is well-defined (independent of the choice of representative μ) because μ represents a class in H1(TXt0(−Dt0 − p0))
and the right-hand side changes by a $\bar\partial$-exact term when μ changes by $\bar\partial$ of a vector field. In
particular, if obκ(s) ≠ 0, then
s cannot persist along the
corresponding base direction; infinitesimally, this means that the
isomonodromic direction determined by κ is to the incidence of sections,
hence transverse to ℬ at (t0, p0).
We next explain why one can choose κ so that is nonzero. By Serre
duality for the pointed curve (Xt0, Dt0 + p0),
we have a perfect pairing
H1(TXt0(−Dt0 − p0)) × H0(ωXt0⊗2(Dt0 + p0)) → ℂ.
On the other hand, pairing the E-valued (1, 0)-form $\widetilde\nabla^{1,0}(s)$ with a
holomorphic section of Ẽt0∨ ⊗ ωXt0(Dt0)
produces a quadratic differential with at most a simple pole at p0. More precisely, for
any
η ∈ H0(Xt0, Ẽt0∨ ⊗ ωXt0(Dt0)),
the contraction $\langle
\eta,\widetilde\nabla^{1,0}(s)\rangle$ defines a section
where the extra +p0
allows for the pole coming from s ∈ Ẽt0(p0).
With these notations, the Serre-dual pairing of the obstruction class
with η may be written (up to
the standard identifications in Dolbeault theory) as
i.e. the obstruction is detected by pairing the Kodaira–Spencer class
with the quadratic differential .
Consequently, to find κ with obκ(s) ≠ 0, it suffices to find η such that qη, s ≠ 0; then we choose κ pairing nontrivially with qη, s under Serre duality, which forces ⟨obκ(s), η⟩ ≠ 0, hence obκ(s) ≠ 0.
The existence of such an η is the point at which irreducibility and nontriviality of the monodromy enter. Indeed, suppose for contradiction that qη, s = 0 for every η ∈ H0(Ẽt0∨ ⊗ ωXt0(Dt0)). Then implies that $\widetilde\nabla^{1,0}(s)$ is annihilated by all holomorphic η, hence vanishes as a holomorphic E-valued (1, 0)-form on the complement of its pole locus. In other words, away from Dt0 ∪ {p0} the section s is locally covariantly constant in type (1, 0). Since s is also holomorphic for the $\bar\partial$-operator defining Ẽt0(p0), it follows that s is flat on Xt0 \ (Dt0 ∪ {p0}) for the induced flat connection. A flat section on the punctured curve determines a nonzero invariant vector in the local system on π1(Xt0 \ Dt0) (the extra puncture at p0 is inessential for monodromy, since the section has at most a simple pole there), contradicting the assumption that ρ is nontrivial and irreducible. Thus there exists at least one η such that qη, s ≠ 0, and we may choose κ with ⟨κ, qη, s⟩ ≠ 0. By , this forces obκ(s) ≠ 0.
We summarize this discussion in the following form, which matches Lemma~3 and Lemma~4 from our list.
In applications one must address the possibility that H0(Ẽt0(p0)) has dimension > 1. Since this space is finite-dimensional, we may choose finitely many nonzero sections s1, …, sm spanning it. Repeating the argument above and using the linearity of κ ↦ obκ(si), we find a single class κ that kills all si simultaneously: one chooses ηi with qηi, si ≠ 0 and then chooses κ pairing nontrivially with each qηi, si. Here the hypothesis g ≥ 3 (and the presence of at least one marked point among Dt0 + p0 in the logarithmic case) ensures that the space H1(TXt0(−Dt0 − p0)) is large enough to avoid a finite union of hyperplanes defined by the vanishing of these pairings. This is the concrete content of the ``killing direction’’ statement (Lemma~4).
We now extract the global consequence. Since ℬ is a closed determinantal analytic subset of 𝒳, if ℬ = 𝒳 then every point (t, p) would admit a nonzero section of Ẽt(p). In particular, at every (t, p) we could choose 0 ≠ s ∈ H0(Ẽt(p)), and Proposition~ would then produce an isomonodromic direction at (t, p) along which s fails to persist. This is incompatible with ℬ being all of 𝒳, because the condition H0(Ẽt(p)) ≠ 0 is closed under specialization but not forced under such transverse deformations. Concretely, Proposition~ shows that through a smooth point of the incidence variety of sections the projection to the deformation space cannot be locally constant; hence ℬ cannot contain an open neighborhood of any of its points. Therefore:
Finally, because ℬ is determinantal
and obtained from algebraic data (after shrinking to an algebraic
neighborhood in moduli), properness in the analytic sense implies
properness in the algebraic sense: no irreducible component of ℬ can coincide with 𝒳, and the image in T of any component of ℬ is a proper closed analytic subset, hence
contained in a proper algebraic subset after shrinking. In particular,
the set of parameters t ∈ T for which the
fiberwise locus
ℬt := { p ∈ Xt ∣ H0(Xt, Ẽt(p)) ≠ 0 }
fails to be a proper subset of Xt is itself a
proper analytic subset of T.
This is the precise ``transversality input’’ needed in the next section:
combined with semicontinuity and the determinantal description of ℬ, it allows us to deduce that for very
general t ∈ T the set
ℬt is a proper
closed subset of the curve Xt, hence H0(Xt, Ẽt(p)) = 0
for a general point p ∈ Xt.
In this section we assume D = ⌀ and we prove Theorem~A, i.e. horizontal generic vanishing for the isomonodromic deformation of an irreducible nontrivial unitary rank-2 local system on a genus g ≥ 3 curve.
Let $(\mathcal X\to T,\widetilde
E,\widetilde\nabla)$ be the isomonodromic deformation over a
local universal deformation space T of X. For (t, p) ∈ 𝒳 we consider the
rank-2 bundle Ẽt(p)
on the fiber Xt, and we
define the Brill–Noether locus
$$
\mathcal B\ :=\ \bigl\{(t,p)\in \mathcal X\ \bigm|\ H^0(X_t,\widetilde
E_t(p))\neq 0\bigr\}.
$$
By semicontinuity of h0 in proper flat
families and the standard determinantal description of the condition
h0 ≥ 1, the set
ℬ is a closed analytic subset of 𝒳 (after shrinking T if necessary). Our goal is to show
that for very general t ∈ T, the fiberwise
locus
ℬt := ℬ ∩ Xt = {p ∈ Xt ∣ H0(Xt, Ẽt(p)) ≠ 0}
is a proper closed subset of Xt; since Xt is a curve,
this is equivalent to ℬt being finite, hence
equivalent to the statement that H0(Xt, Ẽt(p)) = 0
for general p ∈ Xt.
The key input from the previous subsection is Corollary~, which asserts that ℬ ⊂ 𝒳 is . Concretely, there exists at least one point (t1, p1) ∈ 𝒳 such that H0(Xt1, Ẽt1(p1)) = 0, i.e. (t1, p1) ∉ ℬ. We now explain why this already forces the desired generic statement.
Consider the projection π : 𝒳 → T. Since ℬ is closed analytic in 𝒳, for each integer k ≥ 0 the locus
T ≥ k := {t ∈ T ∣ dim (ℬt) ≥ k}
is a closed analytic subset of T by upper semicontinuity of fiber
dimension for analytic maps. Because each fiber Xt is
one-dimensional, the condition that ℬt fails to be a proper
subset of Xt is equivalent
to dim (ℬt) = 1,
i.e. ℬt = Xt
(as ℬt is closed in
the irreducible curve Xt). Thus the
``bad’’ parameter locus is exactly
Tbad := {t ∈ T ∣ ℬt = Xt} = T ≥ 1,
which is a closed analytic subset of T.
We claim that Tbad is a subset of T. Indeed, since (t1, p1) ∉ ℬ, we have p1 ∉ ℬt1, hence ℬt1 ≠ Xt1, and therefore t1 ∉ Tbad. This shows Tbad ≠ T.
Consequently, for all t ∈ T \ Tbad
we have ℬt ⊊ Xt,
hence ℬt is a
finite set. In particular, for any such t, for a general point p ∈ Xt
one has p ∉ ℬt,
i.e.
H0(Xt, Ẽt(p)) = 0.
Since Tbad is a
proper closed analytic subset, its complement is Zariski dense; in
particular the statement holds for very general t ∈ T, which is precisely
the desired horizontal generic vanishing.
We now record the equivalence with generic global generation (still
in the compact case D = ⌀).
For fixed t, by Serre duality
one has
H0(Xt, Ẽt(p))∨ ≃ H1(Xt, Ẽt∨ ⊗ ωXt(−p)),
and for a general point p the
vanishing of H0(Ẽt(p))
is equivalent to the statement that the evaluation map
H0(Xt, Ẽt∨ ⊗ ωXt) → (Ẽt∨ ⊗ ωXt)|p
is surjective. Equivalently, Ẽt∨ ⊗ ωXt
is generated by global sections at the generic point. This is the form
needed in applications to fixed-part vanishing, since global generation
of Ẽt∨ ⊗ ωXt
controls the Kodaira–Spencer contractions appearing in the IVHS
argument.
Finally, we indicate where the hypotheses enter and why they are close to optimal for a degree-1 statement.
If the monodromy representation is trivial, then the underlying
holomorphic bundle is holomorphically trivial, Ẽt ≃ 𝒪Xt⊕2
for all t (indeed, the
isomonodromic deformation is constant as a flat bundle). In this case
Ẽt(p) ≃ 𝒪Xt(p)⊕2
and hence
H0(Xt, Ẽt(p)) ≃ H0(Xt, 𝒪Xt(p))⊕2 ≠ 0
for every p. Thus
nontriviality is necessary.
If ρ is reducible unitary,
then after choosing a unitary decomposition one has a splitting of flat
bundles, and the associated holomorphic bundle decomposes as a direct
sum of degree-0 unitary line bundles,
say Ẽt ≃ Lt ⊕ Lt−1
with Lt ∈ Pic0(Xt)
varying isomonodromically. Then
H0(Xt, Ẽt(p)) ≃ H0(Xt, Lt(p)) ⊕ H0(Xt, Lt−1(p)).
For special choices (e.g. when Lt is trivial,
or more generally when Lt lies in the
theta divisor translate relevant to p), one obtains sections for many
p, and there is no reason for
the Brill–Noether locus to be forced to be proper by the transversality
mechanism: the flat sub-line bundle provides ``too many’’ horizontal
sections. This explains the irreducibility requirement in Proposition~
and ultimately in Theorem~A.
In the transversality argument, after fixing finitely many candidate
sections si ∈ H0(Xt0, Ẽt0(p0)),
one must choose a Kodaira–Spencer class κ avoiding finitely many hyperplanes
in H1(Xt0, TXt0(−p0)).
The dimension of this space is 3g − 4, so for g ≥ 3 it is large enough to ensure
that a finite union of proper hyperplanes cannot exhaust it. In genus
2 this dimension is only 2, and it becomes plausible that the
obstruction hyperplanes attached to different sections (or to different
choices of η detecting those
sections) could fill the deformation space, allowing loci where H0(Ẽt(p)) ≠ 0
persist for all t in a family.
While one can sometimes still prove weaker statements in low genus under
extra hypotheses, the clean very-general vanishing statement of
Theorem~A is naturally aligned with g ≥ 3.
We have thus established horizontal generic vanishing in the compact case. In the next section we extend the argument to the logarithmic setting D ≠ ⌀ by replacing ωXt with ωXt(Dt) and by working with the Deligne canonical extension, i.e. residues in [0, 1), so that the relevant stability and Brill–Noether statements become parabolic. The structure of the proof is unchanged, but one must keep track of the local residue types and, in some cases, perform a finite case analysis depending on the unitary local monodromy eigenvalues.
We now allow D ≠ ⌀ and explain the modifications needed to extend the compact argument to the logarithmic setting, thereby proving Theorem~B. Throughout we keep the hypotheses that ρ : π1(X∘) → U(2) is unitary, irreducible, and nontrivial, and we work with the Deligne canonical extension (E, ∇), i.e. we choose the logarithmic extension for which every residue eigenvalue has real part in [0, 1).
For each marked point xi ∈ D,
the unitary local monodromy Mi ∈ U(2)
is semisimple with eigenvalues on the unit circle. Choosing the Deligne
extension amounts to choosing logarithmic residues Resxi(∇)
whose eigenvalues lie in [0, 1) and
exponentiate to Mi. In rank
2 we may write these eigenvalues
as
0 ≤ αi(1) ≤ αi(2) < 1, spec(Resxi(∇)) = {αi(1), αi(2)}.
This data canonically defines a parabolic structure on the underlying
holomorphic bundle E: at xi we take the
filtration by residue eigenspaces (or generalized eigenspaces, though in
the unitary case there is no nilpotent part), with weights αi(1), αi(2).
Concretely, if αi(1) < αi(2)
then we obtain a full flag
Exi ⊋ Fi ⊋ 0, dim Fi = 1,
with weights αi(1)
on Exi/Fi
and αi(2)
on Fi. If
αi(1) = αi(2)
then the filtration is trivial (no distinguished line Fi), and the
parabolic structure at xi is determined
solely by the common weight.
The point of fixing the Deligne interval is that it makes this parabolic datum canonical and locally constant under isomonodromic deformation: since the monodromy conjugacy classes are fixed along T, the eigenvalues αi(j) ∈ [0, 1) and the corresponding weight system do not jump.
By Mehta–Seshadri, a unitary representation of π1(X∘)
corresponds to a polystable parabolic bundle of parabolic degree 0 on (X, D) with those weights.
Under our irreducibility hypothesis, the associated parabolic bundle is
in fact (rather than merely polystable): any proper parabolic subbundle
would give a nontrivial unitary sub-local system, contradicting
irreducibility. This stability input is the logarithmic substitute for
the stability used implicitly in the compact Brill–Noether discussion,
and it is what prevents sections of Ẽt(p)
from persisting for all complex structures.
One immediate numerical consequence that we use repeatedly is that
the ordinary degree of the Deligne extension satisfies
$$
\deg(E)\ =\ -\sum_{i=1}^n
\mathrm{tr}\bigl(\mathrm{Res}_{x_i}(\nabla)\bigr)
\ \le\ 0,
$$
since each residue eigenvalue lies in [0, 1). In particular deg (E(−D)) < 0,
hence
This vanishing is the logarithmic analogue of the fact that a stable
degree-0 bundle has no nonzero
sections.
Let (𝒳, 𝒟) → T be a local
universal deformation of the pointed curve (X, D) and $(\widetilde E,\widetilde\nabla)$ the
isomonodromic deformation of (E, ∇). The crucial analytic feature
is that, after shrinking T, we
may choose local coordinates near each component of 𝒟 so that $\widetilde\nabla$ has logarithmic
singularities along 𝒟 with residue
endomorphisms constant along T. Thus the parabolic weights and,
in the distinct-eigenvalue case, the parabolic lines Fi ⊂ Exi
are transported horizontally.
For deformations of curves, Kodaira–Spencer classes live in H1(X, TX(−D)).
Equivalently, we may represent a first-order deformation by a Beltrami
differential μ vanishing at
the marked points. This vanishing is exactly what ensures that the
formula from Lemma~1 continues to make sense in the logarithmic setting:
locally, ∇1, 0 has at worst
simple poles along D, and
contracting with a Beltrami differential that vanishes at D produces an L2 (indeed smooth) (0, 1)-form with values in E. In particular, the first-order
deformation of the (parabolic) $\bar\partial$-operator is still given
by
$$
\delta_{\mu}\bar\partial(s)\ =\ \mu\lrcorner \nabla^{1,0}s,
$$
now interpreted in the category of parabolic bundles (or, equivalently,
for the Deligne extension with controlled logarithmic behavior at D). This is the only place where we
truly use that we deform the curve rather than the underlying curve
while keeping D fixed.
We define the Brill–Noether locus ℬ ⊂ 𝒳
exactly as before, with Ẽt(p)
the twist by a point p ∈ Xt
(and we implicitly restrict to p ∉ Dt
when discussing generic behavior). Semicontinuity again shows that ℬ is closed analytic.
The substantive input is the parabolic analogue of the properness statement: one shows that ℬ is a proper subset of 𝒳. Conceptually, nothing changes: if at some base point (t0, p0) we had a nonzero section s ∈ H0(Xt0, Ẽt0(p0)), then Lemma~3 expresses the obstruction to extending s along a Kodaira–Spencer direction κ ∈ H1(Xt0, TXt0(−Dt0)) in terms of the contraction μ⌟∇1, 0s (with μ representing κ). Irreducibility and nontriviality again give Lemma~4: we can choose κ so that the resulting obstruction classes are nonzero simultaneously for a finite spanning set of candidate sections, forcing the Brill–Noether condition to be cut out transversely. Once ℬ is known to be proper, the dimension-theoretic argument on the parameter locus Tbad goes through verbatim, because each fiber Xt remains one-dimensional.
While the global structure of the proof is unchanged, the local analysis
near each xi ∈ D
depends on the , and it is here that one may need to separate finitely
many cases.
Because there are only finitely many punctures and only finitely many residue patterns at each puncture, this results at worst in a finite case analysis. Importantly, the Deligne condition αi(j) ∈ [0, 1) prevents any ambiguity coming from integral shifts of residues, so the case distinction is purely on the multiplicity pattern of eigenvalues (and, in practice, on whether the local monodromy is scalar).
For applications we package the vanishing H0(Xt, Ẽt(p)) = 0
as a generic generation statement for
Ft := Ẽt∨ ⊗ ωXt(Dt).
First, implies H1(Xt, Ft) = 0
for every t, since by Serre
duality
H1(Xt, Ft)∨ ≃ H0(Xt, Ẽt(−Dt)) = 0.
Fix t and let p ∈ Xt \ Dt.
Then the exact sequence
0 → Ft(−p) → Ft → Ft|p → 0
shows that surjectivity of the evaluation map H0(Ft) → Ft|p
is equivalent to H1(Ft(−p)) = 0
(using H1(Ft) = 0).
In the parabolic category, Serre duality takes the form
H1(Xt, Ft(−p))∨ ≃ H0(Xt, Ẽt(p)),
where the appearance of ωXt(Dt)
reflects the logarithmic dualizing sheaf. Since we work with the Deligne
extension (weights in [0, 1)),
parabolic sections coincide with ordinary holomorphic sections of the
underlying bundle, so the right-hand side is exactly the H0 appearing in
Theorem~B. Thus for general p
we have
H0(Xt, Ẽt(p)) = 0 ⇔ H0(Ft) ↠ Ft|p,
i.e. Ft is
generated by global sections at the generic point.
This is the precise form needed in the next section, because the logarithmic Kodaira–Spencer map takes values in H1(TXt(−Dt)), and its contractions naturally pair with global sections of Ẽt∨ ⊗ ωXt(Dt).
We have therefore reduced the logarithmic case to the same two ingredients as in the compact case: properness of the (parabolic) Brill–Noether locus, and the dimension-theoretic deduction of very-general fiberwise finiteness. The only genuinely new features are the bookkeeping of residues/weights via the Deligne interval and the finite residue-type case analysis outlined above. With Theorem~B in hand, we turn next to the fixed-part vanishing application for families dominating ℳg, n.
We now explain how Theorem~B implies the fixed-part vanishing statement with coefficients, i.e. Corollary~C. The argument is standard in variations of Hodge structure and we follow the scheme of (``generic global generation ⇒ no invariants’’).
Let π : (𝒞, 𝒟) → M
be a smooth family of n-pointed curves of genus g ≥ 3 whose classifying map M → ℳg, n
is dominant and generically '{e}tale. Let 𝕌 be a rank-2 unitary local system on 𝒞∘ := 𝒞 \ 𝒟, and assume that for
very general m ∈ M
the fiber representation
ρm : π1(Cm∘) → U(2)
is irreducible and nontrivial. After shrinking M we may assume that this holds for
every m in a dense open
subset, and that the conjugacy classes of local monodromies around the
components of 𝒟 are locally
constant.
Write (ℰ, ∇) for the relative Deligne canonical extension of 𝕌 to 𝒞, i.e. a rank-2 vector bundle on 𝒞 equipped with a flat logarithmic connection with poles along 𝒟 and residue eigenvalues in [0, 1) on each fiber. For each m ∈ M we denote by (Em, ∇m) its restriction to Cm, so that (Em, ∇m) is the Deligne extension attached to ρm.
Consider the relative de Rham complex
$$
\mathrm{DR}_{\mathcal C/M}(\mathcal E)\ :=\ \bigl[\ \mathcal E
\xrightarrow{\ \nabla\ }\ \mathcal E\otimes \omega_{\mathcal
C/M}(\mathcal D)\ \bigr],
$$
placed in degrees 0, 1. By Deligne’s
theory of regular singular connections, the higher direct images
ℋ := R1π*DR𝒞/M(ℰ) and ℍ := R1π*𝕌
are related by ℋ ≃ ℍ⊗ℂ𝒪M, and
ℋ carries the Gauss–Manin connection
∇GM whose flat sections are
precisely the sections of the local system ℍ.
The stupid filtration on DR𝒞/M(ℰ) induces a Hodge
filtration on ℋ. In our situation
(unitary and hence semisimple), the Hodge–to–de Rham spectral sequence
degenerates at E1,
and one identifies the Hodge subbundle
after choosing the standard polarization coming from the unitary metric
(equivalently, passing to the dual system if one prefers to avoid this
identification). Fiberwise this gives
ℱm1 ≃ H0(Cm, Em∨ ⊗ ωCm(Dm)), ℋm ≃ H1(Cm∘, 𝕌m).
We emphasize that is the precise place where the logarithmic dualizing
sheaf ωCm(Dm)
enters.
By Griffiths transversality, the Gauss–Manin connection induces an
𝒪M-linear map on
the associated graded,
θ: ℱ1 → (ℋ/ℱ1) ⊗ ΩM1,
often called the Higgs field or infinitesimal variation of Hodge
structure (IVHS). At a point m ∈ M and a tangent vector
ξ ∈ TM, m,
the component
θm(ξ): ℱm1 → ℋm/ℱm1
is described as a cup product with the Kodaira–Spencer class κ(ξ) ∈ H1(Cm, TCm(−Dm)),
and, under the identifications above, it is computed by contracting
κ(ξ) against
holomorphic Em∨-valued
log-differentials. Concretely, if α ∈ H0(Cm, Em∨ ⊗ ωCm(Dm))
then
where we use the natural pairing TCm(−Dm) ⊗ ωCm(Dm) → 𝒪Cm.
We will use the following standard implication (cf. ): if, for general m, the images of the maps as ξ varies span H1(Cm, Em∨), then ℍ has no nonzero flat sections. The input from Theorem~B is exactly the geometric condition needed to force this spanning property.
Indeed, fix m in the very
general locus where Theorem~B applies to the isomonodromic deformation
of (Em, ∇m)
(equivalently, to the pullback of the universal deformation along the
classifying map M → ℳg, n).
Then Theorem~B gives that the bundle
Fm := Em∨ ⊗ ωCm(Dm)
is generated by global sections at the generic point of Cm. Since the
classifying map is generically '{e}tale, the Kodaira–Spencer map
TM, m → H1(Cm, TCm(−Dm))
is surjective for general m.
Consequently, the maps realize the full family of contractions by
arbitrary Kodaira–Spencer classes.
From generic global generation of Fm we deduce the
needed spanning statement: choose a general point p ∈ Cm \ Dm
such that the evaluation H0(Fm) ↠ (Fm)|p
is surjective. Then for any nonzero class β ∈ H1(Cm, Em∨),
Serre duality gives a nonzero functional
H0(Cm, Em ⊗ ωCm) → ℂ,
and by varying κ ∈ H1(TCm(−Dm))
we can realize, via , enough contractions against global sections of
Fm to
separate β. Equivalently,
there exists κ and α ∈ H0(Fm)
with ⟨β, κ ∪ α⟩ ≠ 0.
This shows that the span of {κ ∪ α} (with κ varying in H1(TCm(−Dm))
and α varying in H0(Fm))
is all of H1(Cm, Em∨).
We now complete the deduction of Corollary~C.
We record one useful representation-theoretic corollary, which is the
form typically used in Prym-type constructions. Suppose one has a family
of (possibly branched) covers varying over M,
f: 𝒴∘ → 𝒞∘,
with Galois group G, and
consider the local system R1f*ℂ
on 𝒞∘. Decomposing the G-representation on cohomology into
isotypic pieces yields a direct sum decomposition of local systems on
𝒞∘ indexed by irreducible
complex representations τ of
G. If for some τ the corresponding factor 𝕌τ has rank 2 and is unitary with irreducible nontrivial
fiber monodromy for very general m, then Corollary~ applied to 𝕌τ shows that
H0(M, R1π*𝕌τ) = 0.
In particular, the monodromy representation of π1(M) on the
τ-isotypic part of the
Prym-type cohomology has no nonzero invariant vectors. Subject to the
usual self-duality constraints (orthogonal versus symplectic, depending
on τ ≃ τ∨
and the polarization), ``no invariants’’ is the first obstruction to the
monodromy group being contained in a proper algebraic subgroup; thus
Corollary~ is a low-rank input forcing largeness of the Zariski closure
on such rank-2 factors.
Finally, we note an interpretation closer to mapping class group dynamics. When M is chosen so that M → ℳg, n is generically '{e}tale, the local system ℍ = R1π*𝕌 is obtained by restricting the tautological cohomology local system on the appropriate character variety (or on Teichm"uller space modulo a subgroup of the mapping class group). The vanishing of H0(M, ℍ) then says that the mapping class orbit of ρ does not carry a nonzero cohomology class fixed under parallel transport along M. In applications, this rules out ``hidden constant directions’’ in families of representations and is frequently the missing hypothesis needed to pass from infinitesimal transversality (provided by Theorem~B) to global rigidity or big-monodromy conclusions.
With these coefficient-vanishing consequences established, we turn next to directions beyond rank 2 and beyond degree-1 twisting.
Theorem~B is a degree-1 statement in the sense that we twist by a single point and obtain horizontal generic vanishing for H0(Et(p)). There are at least three natural directions in which one would like to push the picture: (i) replace p by an effective divisor Z of higher degree, (ii) replace rank 2 by arbitrary rank, and (iii) relate the resulting ``no fixed part’’ consequences to the size of monodromy groups and to arithmetic notions such as integrality and rigidity. We briefly indicate what we expect, what seems approachable by the same method, and where new ideas appear necessary.
Fix d ≥ 1 and let Z ∈ Xt(d)
be an effective divisor of degree d on a fiber. The most naive
extension of Theorem~B would assert that for very general t ∈ T and general Z ∈ Xt(d)
one has
Equivalently, one expects generic global generation properties for the
bundles Ẽt∨ ⊗ ωXt(Dt)
not merely at a general point but in a form compatible with imposing
vanishing at d general points.
Concretely, one may reformulate as the statement that for general Z the evaluation map
H0(Xt, Ẽt∨ ⊗ ωXt(Dt)) → (Ẽt∨ ⊗ ωXt(Dt))|Z
is injective on the annihilator of a suitable subspace, or, more
invariantly, that Ẽt∨ ⊗ ωXt(Dt)
separates d general points in
the expected range. One expects a threshold phenomenon: for a rank-2 degree-0
stable bundle on a general curve, Brill–Noether heuristics suggest that
H0(E(Z))
should vanish for d well below
g, while for d ≫ 0 the space of sections should
become nonzero by Riemann–Roch. Thus, rather than asking for for all
d, a more reasonable goal is
to determine the maximal d = d(g) for which
horizontal generic vanishing still holds.
From the perspective of our proof of Theorem~B, the correct parameter
space is the relative symmetric product 𝒳T(d) → T
and the Brill–Noether-type locus
$$
\mathcal W_d\ :=\ \bigl\{(t,Z)\in T\times X_t^{(d)}\ \bigm|\
H^0(X_t,\widetilde E_t(Z))\neq 0\bigr\}.
$$
Semicontinuity again shows that 𝒲d is analytic, so the
problem reduces to showing that 𝒲d is a proper subset.
The obstruction-theoretic mechanism from Lemma~3 continues to apply: a
nonzero section of E(Z) defines a class in
H0(E(Z)),
and a Kodaira–Spencer direction κ ∈ H1(TX(−D))
produces a first-order obstruction in H1(E(Z))
computed by the same contraction formula as in Lemma~1, now with poles
permitted along Z as well.
What changes is that the space of potential sections can grow with d, so one must control
simultaneously a larger family of sections and the dependence on the
moving divisor Z.
Two technical issues appear at this point. First, the deformation problem now mixes two types of variation: variation of the complex structure (horizontal direction) and variation of Z inside Xt(d). In the degree-1 case we could fix p general and focus on horizontal variation; for general d we must keep track of how moving Z may compensate for a killing direction in T. Second, even if one proves that for each fixed (t, Z) there exists a killing direction, one needs uniformity in families to deduce that the locus is not dominant over T. A plausible strategy is to show that the differential of the ``section existence’’ condition has maximal rank at some point, producing codimension estimates for 𝒲d. Such a statement would be a genuinely quantitative refinement of Theorem~B.
For a unitary representation ρ : π1(X∘) → U(r) with r > 2, one can still form the Deligne canonical extension (E, ∇) and its isomonodromic deformation. The first-variation formula of Lemma~1 is rank-independent: for a fixed unitary connection on the underlying smooth bundle, the variation of $\bar\partial$ in a Beltrami direction is given by contraction with ∇1, 0. Thus the deformation-theoretic input that makes the argument ``horizontal’’ survives unchanged.
The place where rank 2 was used more seriously is in the transversality step: we needed a mechanism guaranteeing that, given a nonzero would-be section of E(p), there exists some Kodaira–Spencer class κ that obstructs it, and moreover that one can simultaneously obstruct all sections that might appear. In rank 2 we can exploit irreducibility and nontriviality to rule out the presence of too many ∇-parallel subobjects, and we may invoke stability of the underlying (parabolic) bundle for very general complex structure. In higher rank, the space of potential destabilizing subbundles is larger and the Brill–Noether loci for H0(E(p)) ≠ 0 may have more components with different geometric meanings (coming, for example, from special subrepresentations after restriction to subsurfaces). Accordingly, we view the following as a natural target.
A weaker intermediate statement, which may be closer to the current method, is to prove generic global generation after passing to suitable Schur functors. For instance, one might attempt to show that ∧r − 1Ẽt∨ ⊗ ωXt(Dt) is generically globally generated, which is equivalent to vanishing for Ẽt(p) after dualizing and twisting by det (Ẽt). Since unitary local systems have degree 0 and det (Ẽt) is flat, such reductions may sometimes lower the problem to controlling line subbundles in exterior powers, where stability arguments are more classical.
Corollary~C asserts that for families dominating ℳg, n, the local
system R1π*𝕌
has no nonzero global sections, i.e. no π1(M)-invariants.
In concrete applications one typically wants more: one wants the
identity component of the Zariski closure of the monodromy
representation
π1(M) → GL(H1(Cm∘, 𝕌m))
to be as large as allowed by polarization and self-duality constraints.
The absence of invariants is merely the first obstruction to monodromy
being contained in a proper subgroup.
One direction, suggested by Prym-type constructions, is to combine Corollary~C with representation-theoretic ``bootstrapping’’: if we know that there are no invariants in a collection of tensor constructions on H1(Cm∘, 𝕌m) (symmetric squares, exterior squares, endomorphism representations), then one can often force the derived group of the Zariski closure to contain a large classical subgroup. Thus it is natural to seek horizontal generic vanishing statements not only for Ẽt(p) but also for twists of bundles naturally attached to 𝕌, such as End(Ẽt) or Sym2(Ẽt). Such statements would feed into vanishing of fixed parts in higher tensor powers of R1π*𝕌 and could be used to rule out monodromy contained in, say, a Levi subgroup.
A complementary direction is dynamical: if one views the isomonodromy foliation on a character variety, Corollary~C says that along a generically immersive base M the pulled-back cohomology local system has no constant vectors. It is natural to ask whether the same methods can prove that there are no constant of small rank, i.e. that R1π*𝕌 contains no nontrivial unitary direct factor. This would be closer in spirit to ``simplicity’’ of monodromy rather than merely absence of invariants, and would require controlling Higgs-field images on higher Grassmannians.
Finally, we comment on possible arithmetic interactions. A recurring theme in the theory of complex local systems is that integrality constraints (for instance, traces in $\overline{\mathbb Q}$, or an underlying ℤ-structure) tend to force strong rigidity or finiteness phenomena, especially in the unitary case. For example, if a unitary representation has eigenvalues that are algebraic integers of absolute value 1, Kronecker-type arguments often imply that these eigenvalues are roots of unity; in favorable situations this forces finite image. Thus, if one imposes strong integrality hypotheses on an irreducible unitary ρ, one expects severe restrictions, potentially contradicting the ``genericity’’ conclusions of Theorem~B except in the finite-image case.
This suggests two concrete problems. First, one may ask for a precise dichotomy: under what arithmetic conditions on ρ (e.g. definability over a number field with bounded denominators) must ρ have finite image, and hence fall into the excluded ``trivial/reducible’’ patterns where horizontal generic vanishing fails? Second, one may seek a geometric criterion for arithmeticity phrased in terms of the variation of the associated holomorphic bundles: for instance, if along a family M the bundles Em exhibit unusually large Brill–Noether loci (persistent sections of Em(Z) for many Z), can one deduce that ρm lies in a special (possibly arithmetic) subvariety of the character variety?
Related to this is the philosophy that cohomological rigidity of a local system should be detectable by vanishing of deformation spaces. While our results are in a different direction (they show that certain cohomology groups contain fixed vectors under deformation of the curve), it is plausible that combining horizontal vanishing with additional constraints (e.g. fixing the conjugacy classes at punctures and imposing motivic or integral conditions) could contribute to proving that certain unitary systems cannot occur in nontrivial families over ℳg, n, or conversely that the only such systems are those coming from finite monodromy.
To summarize, the method underlying Theorem~B appears robust enough to address at least the following refinements with additional work: (a) codimension estimates for the loci where H0(Ẽt(Z)) ≠ 0 in T × Xt(d), yielding quantitative versions of horizontal generic vanishing for small d; (b) extensions to higher rank provided one can supply a suitable transversality lemma replacing Lemma~4; and (c) strengthened fixed-part vanishing for tensor constructions on R1π*𝕌, which would be directly applicable to big-monodromy problems.
On the other hand, genuine arithmetic applications will likely require input beyond deformation theory on curves, namely a bridge between the analytic genericity of the underlying holomorphic bundle and arithmetic constraints on the representation. Establishing such a bridge would place horizontal generic vanishing within the broader landscape of special subvarieties of character varieties and would clarify to what extent unitary isomonodromy can coexist with integrality.