Uniformizing VHS, Anosov reps., Lyapunov exponents]Uniformization of some weight $3$ variations of Hodge structure, Anosov representations, and Lyapunov exponents

Revised August 27, 2024

Simion Filip

Department of Mathematics

University of Chicago

5734 S University Ave

Chicago, IL 60637

We develop a class of uniformizations for certain weight $3$ variations of Hodge structure (VHS).
The analytic properties of the VHS are used to establish a conjecture of Eskin, Kontsevich, Möller, and Zorich on Lyapunov exponents.
Additionally, we prove that the monodromy representations are log-Anosov, a dynamical property that has a number of global consequences for the VHS.
We establish a strong Torelli theorem for the VHS and describe appropriate domains of discontinuity.

Additionally, we classify the hypergeometric differential equations that satisfy our assumptions.
We obtain several multi-parameter families of equations, which include the mirror quintic as well as the six other thin cases of Doran–Morgan and Brav–Thomas.

§ INTRODUCTION

The global geometry of higher weight variations of Hodge structure remains mysterious.
In this paper we investigate one of the simplest nonclassical situations, with Hodge numbers $(1,1,1,1)$, subject to a nonvanishing condition on a component of the second fundamental form.
This covers for instance the case of the mirror quintic, or Dwork, family as well as six other families among the fourteen identified by Doran and Morgan [24] that were later proved to have thin monodromy by Brav–Thomas [14].
As we show below, even in the class of hypergeometric local systems, a much larger class consisting of several multi-parameter families is covered by our analysis (a one-parameter subfamily was analyzed by completely different methods in [30]).

The interest in the geometry of these variations of Hodge structure (VHS for short) comes from many directions, including mirror symmetry, see [16] for the stunning application of the mirror quintic VHS to count rational curves on the generic quintic $3$-fold.
The original motivation of the present work was a conjecture of Eskin, Kontsevich, Möller, and Zorich [26] predicting a formula for the sum of Lyapunov exponents in terms of degrees of Hodge bundles.
This formula was discovered to hold numerically by Kontsevich in seven of the fourteen families from [24].
Around the same time it was discovered by Brav and Thomas that the same seven families had thin monodromy, while Singh and Venkataramana [63] showed the monodromy in the remaining seven surjects onto a lattice in $\Sp_4(\bZ)$.

In this paper we establish the conjectured formula for Lyapunov exponents stated in [26] and in fact several other properties of a VHS which satisfies “assumption A” as per <ref> below.
In short, it requires that the second fundamental form $\sigma_{2,1}\colon \cV^{2,1}\to \cV^{1,2}\otimes \Omega^1_X$ is an isomorphism (and at the cusps it is an isomorphism if we use the logarithmic cotangent bundle).
Geometrically, this condition can be viewed as a generalization of the “tiling” condition for a Fuchsian triangle group, which requires the angles to be of the form $\pi/N$.

We show that the monodromy representation is in fact an Anosov representation, a concept introduced by Labourie [53], except that the image contains unipotent elements and this leads to what we call a “log-Anosov representation”.
Basic facts on this class of representations were also recently developed in the greater generality of relatively hyperbolic groups by Zhu [68] and Kapovich–Leeb [47] under the name relatively dominated, or relatively Anosov, representations.
In the case of Fuchsian groups, an equivalent notion was studied by Canary–Zhang–Zimmer [22] under the name of cusped Anosov representations.

Associated to Anosov, or log-Anosov, representations are domains of discontinuity in flag manifolds.
Our main results show that some of these domains of discontinuity have natural interpretations in terms of Hodge theory.

Two points are of interest.
First, the domains of discontinuity are “nonclassical” in the sense that they are not of the form $G/M$ where $M$ is a compact subgroup of the semisimple Lie group $G$.
In fact, lattices in $G$ will not have domains of discontinuity of the type constructed below.
Second, it is well-known that the Griffiths transversality condition imposes severe restrictions on the dimension of the image of the period map in the Griffiths period domain $\cD$ (which incidentally is of the form $G/M$ for $M$ compact).
In particular, most points in $\cD$ (a full measure set and with Baire-negligible complement) are not in the image of any period map.
It might therefore be of interest that linear-algebraic data in other kinds of domains, described below, can be put into bijective correspondence with data coming from the VHS.

As a consequence of our methods we obtain a strong Torelli theorem for the VHS under consideration, using a pseudo-hermitian period domain parametrizing the Griffiths intermediate Jacobians.
Further uniformization results are described below.

As for dynamical results on log-Anosov representations, we introduce the notion of “stable point” for a general closed subgroup of $\SL(V)$ acting on $\bP(V)$ and use this to deduce the standard proper discontinuity criteria.
We also obtain a minimality result for the action on a limit set which might be of interest because the analogous statement does not hold for Hitchin representations.
Additionally, we can classify the points with rational coordinates that occur on the limit set.

Include explicit nonvanishing power series

§.§ Main results

Let $X$ be a complete finite volume Riemann surface and $\bV\to X$ a variation of Hodge structure with Hodge numbers $(1,1,1,1)$ satisfying assumption A from <ref>.
Let also $\bW:=\Lambda^2_{\circ}\bV$ be the reduced second exterior power, with the line corresponding to the invariant symplectic form removed, so it is a VHS with Hodge numbers $(1,1,1,1,1)$.
Fix a basepoint $x_0\in X$ (omitted most of the time from the notation) and let $\rho\colon \pi_1(X)\to \Sp_4(V_\bR)$ be the monodromy representation, where $V_{\bR}$ is the fiber of $\bV$ over the basepoint, and similarly for $\SO_{2,3}(W_{\bR})$ and $\bW$.

Our first result settles the conjectures made by Eskin, Kontsevich, Möller, and Zorich in [26].

[Formula for the sum of Lyapunov exponents]

With the above assumptions let $\lambda_1\geq \lambda_2\geq 0$ be the nonnegative Lyapunov exponents of the geodesic flow on $\bV$.

\[
\lambda_1 + \lambda_2 = \frac{\deg \cV^{0,3}_{ext} + \deg \cV^{1,2}_{ext}}{\chi(X)}
\]

where $\chi(X)$ is the (orbifold) Euler characteristic of $X$, $\deg$ denotes the (orbifold) degree of a complex line bundle, and the subscript $ext$ denotes the Deligne extension of a bundle across punctures.

Furthermore, for a vector $w\in W_{\bR}$, the “bad locus” (see <cit.>) is empty if $[w]\in \LGr(V_{\bR})\setminus \Omega_L$, where $\Omega_L$ is the domain of discontinuity from <ref>(i).
For a vector in $\Omega_L$ or $\bS^{1,3}(W_{\bR})$, the bad locus consists of precisely one point.

In the hypergeometric examples, the MUM Lagrangian belongs to the complement of $\Omega_L$, so the nonvanishing Conjecture 6.4 of [26] holds.

The reader interested only in the proof of the formula for the sum of Lyapunov exponents from <ref> can read just <ref>.
The rest of the paper is devoted to deducing the Anosov-type properties of the monodromy, as well as analyzing further the domains of discontinuity.

Let $\wtilde{X}$ be the universal cover associated to the basepoint $x_0$, whose lift to $\wtilde{X}$ we will continue to denote by $x_0$.
Equipped with the complete hyperbolic metric, $\wtilde{X}$ can be identified with the hyperbolic plane $\bH^2$ and we will write $\partial \wtilde{X}$ for its boundary, which can be identified with $\bP^1(\bR)$.

[log-Anosov property]

The monodromy representation $\rho$ is log-Anosov, in the sense of <ref>.
Specifically, there exist constants $c,\ve>0$ such that

\[
\mu_1(\rho(\gamma))-\mu_2(\rho(\gamma))\geq \ve \cdot \dist(x_0,\gamma x_0) - c
\]

where $\mu_1\geq \mu_2 \geq 0$ are the logarithms of the singular values and $\dist$ refers to the hyperbolic distance.

Furthermore, there exists a continuous $\rho$-equivariant map

\[
\xi \colon \partial \wtilde{X}\to \bP(V_{\bR})
\]

that is dynamics-preserving and transversal.

Transversality means that for two distinct boundary points $p_1\neq p_2$ the line $\xi(p_1)$ is disjoint from the symplectic-orthogonal of $\xi(p_2)$,
while dynamics-preserving is detailed in <ref>.
Informally, it means that points on the boundary fixed by $\gamma\in \pi_1(X)$ map to points in $\bP(V_{\bR})$ fixed by $\rho(\gamma)$.

<ref> is a concatenation of <ref>, which establishes the singular value gap using Hodge-theoretic techniques, and <ref> which gives the boundary map.
Recall also that the singular values of a matrix $g$ are the square roots of the eigenvalues of $g\cdot g^{\dag}$, and in the case of symplectic matrices the singular values have a symmetry under $\lambda\mapsto \tfrac 1\lambda$, so in the statement of the theorem we only consider the logarithms which yield positive numbers.

A standard consequence of the log-Anosov property is the existence of a generous supply of domains of discontinuity.
Recall that the real Lagrangian Grassmannian $\LGr(V_\bR)$ is the quadric of null vectors in $\bP(W_{\bR})$, were $W=\Lambda^2_{\circ}V$ is the reduced second exterior power.

[Domains of Discontinuity]

Let $\Gamma\subset \Sp_4(V_{\bR})$ be the image of the monodromy group.

* In the real Lagrangian Grassmannian $\LGr(V_\bR)$ there exists an open nonempty set $\Omega_L$ on which $\Gamma$ acts properly discontinuously.

* The pseudosphere $\bS^{1,3}$ of unit vectors in $W_{\bR}$ is a domain of discontinuity for $\Gamma$.

* The complex Grassmannian of Lagrangians of signature $(1,1)$ for the indefinite hermitian metric on $V_{\bC}$, denoted $\LGr^{1,1}(V_{\bC})$, is also a domain of discontinuity for $\Gamma$.

* In the complex projective space $\bP(V_{\bC})$ there exists an open, nonempty set $\Omega_P$ on which $\Gamma$ acts properly discontinuously.

These results are proved in <ref>, <ref>, <ref>, and <ref> respectively.
Note that the log-Anosov property, as well as the existence of the above domains of discontinuity, is a topological, or coarse-geometric future of the monodromy group.
We can show, however, that some of the domains of discontinuity are uniformized using the variation of Hodge structure.

Domains of discontinuity of this flavor were constructed by Guichard–Wienhard <cit.>, see also Benoist [5] who considered domains of discontinuity that are homogeneous under the action of the ambient Lie group.
We provide a self-contained treatment in <ref> in terms of the notion of “stable point”, though the mechanism behind the proof of the proper discontinuity criterion is essentially the same.

§.§.§ Uniformization from Hodge structures

To state the uniformization results, we need a linear-algebraic construction.
Let $F^2 = \cV^{3,0}\oplus \cV^{2,1}$ be the middle term of the Hodge filtration.
To such a complex Lagrangian associate the set of real Lagrangians $L$ such that their complexification is not transverse to $F^2$, i.e. $L_{\bC}\cap F^2\neq 0$.
We denote this set by $\beta(F^2)\subset \LGr(V_{\bR})$ and call it an “electron” (see <ref>), it is a real $1$-dimensional circle in $\LGr(V_{\bR})$ which is moreover timelike for the causal structure on $\LGr(V_{\bR})$.
This allows us to uniformize the domain of discontinuity $\Omega_L\subset \LGr(V_{\bR})$:

[Uniformization of real Lagrangians]

Let $Bad\to X$ be the circle bundle of electrons over $X$, and $\wtilde{Bad}$ its lift to the universal cover $\wtilde{X}$.
Then the natural developing map $\wtilde{Bad}\xrightarrow{\Dev}\LGr(V_{\bR})$ is a diffeomorphism between $\wtilde{Bad}$ and the domain of discontinuity $\Omega_L$.

This is a combination of <ref> and <ref>(ii).
For why the circle bundle is called “bad”, see <ref>.

There is an extension of the uniformization result to the real pseudosphere $\bS^{1,3}(W_{\bR})$.
Recall that $\bW=\left(\Lambda^2_{\circ}\bV\right)$ is the variation of Hodge structure associated to the reduced second exterior power of $\bV$, Tate-twisted from weight $6$ to weight $4$, with Hodge numbers $(1,1,1,1,1)$, and Hodge bundles denoted $\cW^{p,q}$.

[Uniformization of pseudosphere]

Let $\cW^{1,3}_{\times}\to X$ be the $\bC^{\times}$-bundle of nonzero vectors in the Hodge bundle $\cW^{1,3}$, and let $\wtilde{\cW}^{1,3}_{\times}$ be its lift to the universal cover $\wtilde{X}$.

There is a natural developing map $\wtilde{\cW}^{1,3}_{\times}\xrightarrow{\Dev}\bS^{1,3}(W_{\bR})$ which is a bijective diffeomorphism.

The developing map is constructed in <ref>, using the middle Hodge bundle $\cW^{2,2}$ as well as $\cW^{1,3}$, to parametrize the unit vectors orthogonal to $\cW^{0,4}$.

Additionally, in <ref>, we show that if we consider the adjoint representation VHS, then the map from the universal cover $\wtilde{X}$ to the Grassmannian of $2$-planes $\Gr(2;W_{\bR})$ obtained from $\cW^{4,0}\oplus \cW^{0,4}$, has well-defined boundary values.
These coincide with the limit curve $\xi$ from <ref>.

§.§.§ Hypergeometric equations

A large class of examples of VHS satisfying assumption A comes from hypergeometric ordinary differential equations (see <ref> for the relevant notions).
A classification of the hypergeometric parameters which satisfy it is provided by <ref> and reads:

[Good hypergeometric parameters]

The parameters of hypergeometric differential equations that satisfy assumption A are listed in <ref> and <ref>.

<ref> includes the seven thin cases of Brav–Thomas [14], as well as the infinite family from [30].
In particular, assumption A distinguishes the seven thin cases of Brav–Thomas from the remaining seven cases in the Doran–Morgan list.
In the case of an orbifold point of order $N$, with local exponents $0<\alpha_1<\alpha_2<\alpha_3<\alpha_4<1$, the requirement of assumption A is that $N\cdot (\alpha_3-\alpha_2)=1$.
It means that on the finite cover removing the orbifold point, the corresponding second fundamental form is an isomorphism.
Using that $\alpha_3=1-\alpha_2$ it follows that $\alpha_2=\frac{N-1}{2N}$.
A glance at the table from [14] reveals that this is precisely what distinguishes the thin cases with orbifold points from the remaining cases.
Note that four of the thin cases have cusps instead of orbifold points, where assumption A is analyzed in <ref>.

$\alpha_{\bullet}$                                         $\beta_{\bullet}$                                  conditions

$\left(\mu,\tfrac 12,\tfrac 12,1-\mu\right)$     $(0,0,0,0)$      $\mu\in\left(0,\tfrac12\right]$

$\left(0,0,\nu,1-\nu\right)$     $0<\nu<\mu \leq\tfrac 12$

Parameters $\mu,\nu$ are real.

$\beta_\bullet$      conditions

\tfrac 12,\tfrac 12,\tfrac 12,\tfrac 12
\right)$

$\left(0,0,\mu,1\!-\!\mu\right)$     $0<\mu<\frac{2k-1}{2N}$

\mu,\tfrac 12,\tfrac 12,1\!-\!\mu
\right)$                             $\tfrac{N-1}{2N}<\mu<\tfrac 12$

$ \left(
\tfrac{M-(2k_M+1)}{2M},
\tfrac{M-1}{2M},
\tfrac{M+1}{2M},
\tfrac{M+(2k_M+1)}{2M}
\right)$                             $\tfrac{2k_M+1}{M} < \tfrac 1N$

Parameters $M,k_M$ are integers, $\mu$ is real.

Throughout $\alpha_{\bullet}= \left(
\tfrac{N-(2k+1)}{2N},
\tfrac{N-1}{2N},
\tfrac{N+1}{2N},
\tfrac{N+(2k+1)}{2N}
\right)$
with arbitrary integers $k\geq 1, N>2k+1$.

The tables are constructed from two restrictions.
One is local, and specifies what the exponents of the ODE have to be in order for the second fundamental form $\sigma_{2,1}$ to be an isomorphism (see <ref>).
The second restriction is global and comes from results of Fedorov [29], and requires the exponents $\alpha_\bullet,\beta_\bullet$ to be arranged in a specific way $\mod 1$, so that the Hodge numbers are $(1,1,1,1)$, see <ref>.

Let us note that the families above are of several types (in all cases subject to some inequalities on parameters): one type of family depends on four integers $M,N,k_M,k_N$, another type of family depends on two real numbers $\mu,\nu$, and finally there are mixed cases where the family depends on two integers $(N,k_N)$ and a real number $\mu$.

§.§.§ Torelli theorems

With the above results, we can gain some insight into the global structure of some moduli spaces of Calabi–Yau $3$-folds.
Specifically, the real uniformization result from <ref> implies a strong Torelli theorem for VHS satisfying assumption A, in particular for the much-studied mirror quintic introduced in [16] and the six other families from [24] satisfying our assumption:

[Strong Torelli using Lagrangians]

Suppose that $\bV\to X$ is a VHS satisfying assumption A, and lift it to the universal cover $\wtilde{X}$.

* The maps induced by the Hodge filtration $F^2=\cV^{3,0}\oplus \cV^{2,1}$:

\begin{align*}
\wtilde{X}&\xrightarrow{F^2} \LGr^{1,1}(V_{\bC})\\
\intertext{and taking the quotient by $\pi_1$:}
{X}&\xrightarrow{F^2} \leftquot{\Gamma}{\LGr^{1,1}(V_{\bC})}\\
\end{align*}

are injective.

* Furthermore, for $x,y\in \wtilde{X}$, if $L$ is a real Lagrangian such that $\left(F^2(x)\cap L_{\bC}\right)\neq 0 \neq \left(F^2(y)\cap L_{\bC}\right)$ then $x=y$.

A further dichotomy, related to rational Lagrangian subspaces, is contained in <ref>.

The claims are an immediate consequence of <ref>, which associates to $F^2$ the electron $\beta(F^2)$ and establishes injectivity of the corresponding developing map.

For the mirror quintic family, a generic Torelli theorem using the full Griffiths period domain was proved by Usui [65].

[On Griffiths intermediate Jacobians]

When the real weight $3$ Hodge structure has an underlying integral structure, one can associate to it the Griffiths intermediate Jacobian $V_{\bZ}\backslash V_{\bC}/F^2$.
In general this is not an abelian variety, and the period domain of such objects is $\LGr^{1,1}(V_{\bC})$, a pseudo-hermitian homogeneous space, in contrast to Siegel spaces parametrizing marked abelian varieties.

It is not possible to take a Hausdorff quotient of $\LGr^{1,1}(V_{\bC})$ by $\Sp_4(\bZ)$, or any lattice in $\Sp_4(\bR)$.
However, <ref> implies that for the monodromy $\Gamma$ of a VHS satisfying assumption A, it is possible to take the quotient.
<ref> then says that the period map to this quotient is injective.
So it can be viewed a Torelli theorem in the classical sense.

§.§.§ Integral vectors

The “limit set” curves provided by <ref> are fractal curves (in fact, it is possible and not hard to prove that in hypergeometric examples, because of the rank $1$ unipotent, the limit curve cannot be rectifiable).
expand on lack of rectifiability after showing minimality
Nonetheless, we can classify the rational points on the curve, and similarly for the limit set in the Lagrangian Grassmannian.
This is possible when the monodromy is contained in integer matrices, for instance in the mirror quintic example as well as the six other families of Doran–Morgan [24] that satisfy assumption A.

[Rational directions on limit curve]

Suppose that the image of the monodromy group $\Gamma$ is contained in $\Sp_4(\bZ)$.
Let $\xi\colon \partial\wtilde{X}\to \bP(V_\bR)$ be the boundary map, and let $\Lambda_L\subset \LGr(V_{\bR})$ be defined as the complement of the domain of discontinuity $\Omega_L\subset \LGr(V_{\bR})$ from <ref>.

* A line $[v]\in \xi(\partial\wtilde{X})\subset \bP(V_\bR)$ has rational coordinates if and only if there exists a unipotent transformation $\gamma\in \Gamma$ such that $v$ is both in the kernel and image of $\gamma-\id$.
In particular, the rational vectors on $\xi(\partial\wtilde{X})$ fall into finitely many orbits under the action of $\Gamma$, corresponding to the cusps of $X$.

* A line $[w]\in \LGr\left(V_\bQ\right)$ representing a Lagrangian $L_{[w]}\subset V_{\bQ}$ is contained in $\Lambda_L$ if and only if there exists a unipotent $\gamma\in \Gamma$ and $[v]$ as in part (i), fixed by $\gamma$, such that $[v]\subset L_{[w]}$.

This result follows from <ref>, which establishes the result in slightly greater generality.
This result should be useful to build a compactification of the quotients of some of the domains of discontinuity appearing in <ref>, particularly $\Omega_L\subset \LGr(V_\bR)$.

<ref> above, combined with the strong Torelli theorem provides an interesting property of rational Lagrangian subspaces:
[Dichotomy for rational Lagrangian subspaces]

With assumptions as in <ref>, for every rational Lagrangian subspace $L\subset V_{\bQ}$, precisely one of the following holds:

* either there exists a unipotent transformation $\gamma\in \Gamma$ and vector $v\in L$ such that $v$ is both in the kernel and in the image of $\gamma-\id$,

* or there exists a unique $x\in \wtilde{X}$ such that $L_{\bC}\cap F^2(x)\neq \{0\}$.

Note that the property of being both in the kernel, and in the image, of $\gamma-\id$ is equivalent (in $\Sp_4$) to $v$ belonging to the deepest part of the monodromy weight filtration, see <ref>.

§.§.§ Minimality of the action

In the case of Hitchin representations (see [45] were they were introduced) to $\Sp_{4}(\bR)$, there is a limit “curve” $\Lambda_P\subset \bP(V_{\bR})$ and a limit surface $\Lambda_L\subset \LGr(V_{\bR})$, but furthermore there is a continuous equivariant map $\Lambda_P\into \Lambda_L$, in other words there is an equivariant choice of full flag.
The action of the group on the limit surface $\Lambda_L$ preserves the limit curve, and on the complement the action is conjugated to that of a Fuchsian lattice in $\SL_{2}(\bR)$ acting on $\SL_{2}(\bR)/A$, where $A$ is the diagonal group.
In particular, the orbits of the action are “wild” and there are orbit closures of arbitrary Hausdorff dimension below $2$.

It is therefore of interest that the following holds:
[Minimality on limit set]

Suppose $\rho\colon \pi_1(X)\to \Sp_4(\bR)$ is an $\alpha_1$-log-Anosov representation as above.
Suppose also that there is at least one parabolic boundary point where the monodromy is a rank $1$ unipotent.

Then the action on the limit set $\Lambda_L\subset \LGr(V_{\bR})$ is minimal.

This result is established in <ref>.
Note that all hypergeometric examples of <ref> satisfy the assumptions.
In particular, the equivariant measurable section provided by the Oseledets theorem has dense image in $\Lambda_L$ (see <ref>).

§.§.§ Schwarz reflection in higher rank

Hypergeometric equations with real parameters have an additional complex-conjugation symmetry.
This allows one to embed with index $2$ the monodromy group in a group generated by three involutions, corresponding to the three segments on the real projective line formed by eliminating $0,1,\infty$.
In fact, explicit formulas à la Levelt are possible, see <ref>.

This information can be used to “Schwarz-reflect” the variation of Hodge structure, in analogy with the case of triangle reflection groups.
First, recall that in the case of triangle reflection groups, the image of the period map on the real axis is necessarily a subsegment of a hyperbolic geodesic, and the geodesic itself is determined by the “monodromy” reflection which fixes it.

Two notable features arise in this higher weight situation.
First, the image of the period map is not uniquely determined by the monodromy reflection.
However, the Griffiths transversality conditions pleasantly decouple and the image is essentially determined by a spacelike curve in the pseudo-Riemannian space $\bS^{1,1}$.
This puts rather stringent conditions on its global behavior, and in particular makes it the graph of a globally Lipschitz function in appropriate coordinates.

The second feature is the construction of a tiling of the domain of discontinuity $\bS^{1,3}(W_\bR)$ from <ref>, using polyhedra obtained from the variation of Hodge structure.
We refer to <ref> for more details on this construction and remark that this tiling can be used as an alternative route to obtaining the log-Anosov property and its consequences.
Our assumption A can be understood geometrically as a tiling condition, analogous to the one arising in the polyhedron theorem.
It seems likely that this approach can be generalized to higher rank hypergeometric variations.

§.§ Further techniques and results

§.§.§ GIT and proper discontinuity

There is a large literature on criteria for proper discontinuity of an action of a closed subgroup $\Gamma$ in a semisimple Lie group $G$ on a $G$-space $M$.
For example Benoist [5] considers $M=G/H$ for noncompact subgroups $H$, and related criteria for domains in flag manifolds appear in Guichard–Wienhard [44], Kapovich–Leeb–Porti [49], and references therein.
To make the text self-contained, and to provide some tools for more general situations, we consider a general $G$-representation $V$ and show that $\Gamma$ acts properly discontinuously on the set of stable points in $\bP(V)$, see <ref> for a more general statement.
That there is a relation to Mumford's constructions in Geometric Invariant Theory [57] was first suggested by Kapovich–Leeb–Porti [49], who used this point of view to construct domains of discontinuity in flag manifolds.

We take this point of view a bit further and define stable points in an arbitrary $G$-representation $V$ so that the assertion of the Hilbert–Mumford numerical criterion holds (see <ref>).
Semistable points are defined analogously.
The class of $1$-parameter semigroups that we use for the “numerical criterion” are given by the limit set of the group $\Gamma$.
We refer to <ref> for the relevant notions.

§.§.§ Index estimates, variational methods

The proofs of the main analytic properties of the VHS, contained in <ref>, are based on some inequalities which provide control on the growth of vectors under parallel transport.
Some of the arguments have a variational flavor, and indeed it is quite convenient to invoke at certain steps the Palais–Smale condition and the Ekeland variational principle, even though our situation is essentially $2$-dimensional.

§.§.§ Dominated cocycles and posets

While the notion of Anosov, or log-Anosov, representation, is intrinsic to the group $G$, it is almost always necessary to look at particular $G$-representations.
To each $G$-representation $\phi\colon G\to \GL(V)$ there is an associated local system $\bV$, and if there is a VHS on one such local system, it can be propagated to other $G$-representations.

The dynamical properties of the cocycle induced by $\bV$ under parallel transport along the geodesic flow are encoded by the notion of dominated bundle, or cocycle.
In the context of Anosov representations, this was observed by Bochi, Potrie, and Sambarino [13], where most of the analysis was concerned with cocycles with only two pieces, one strictly dominating the other.
The natural structure, in the context of Anosov representations, is a splitting indexed by a poset, see <ref>.
This allows us to do a finer analysis of the dynamical behavior.
We illustrate the flexibility of this notion with some examples in <ref>.
The necessity to consider poset-indexed bundles in domination arises when considering representations other than the “standard ones”; the adjoint representation of $\Sp_4(\bR)$ for instance exhibits a nontrivial poset structure, see <ref> and <ref> for the general version.
Let us note that poset structures on Lyapunov bundles have also recently appeared in the work of Ledrappier and Lessa [54].

In most situations in the literature, the representation of $G$ is used as an auxiliary tool and can be picked at will, c.f. for instance the “Tits”, or “” representation used in <cit.>.
In the present paper, the representations to consider are dictated by Hodge theory and so we need to extract further dynamical information, encoded by the poset of dominated bundles.

§.§.§ Diffeomorphism types of real domains of discontinuity

The real $3$-manifold obtained as a quotient of the domain of discontinuity in the Lagrangian Grassmannian, constructed in <ref>, has a natural structure of (orbifold) circle bundle over a Riemann surface, by <ref>.
This makes it a Seifert fibered $3$-manifold, for which we refer to the lecture notes [46] for more information.
Furthermore, <ref> provides a uniformization of the corresponding $\bC^{\times}$-bundle by the pseudo-Riemannian sphere $\bS^{1,3}$.
This ties in with Neumann's appendix to [46], which identifies holomorphic $\bC^\times$-bundles (with negative Euler class) with quasi-homogeneous surface singularities and endows the associated circle bundle with a geometric structure of Thurston type (see also [52]).
It would be interesting to understand if there is a relation between the uniformization constructed here using Hodge theory, and those constructed by Neumann.

§.§.§ Thin hypergeometric groups

No lattice in $\Sp_4(\bR)$ can have a domain of discontinuity as in <ref>.
It follows that the monodromy group $\Gamma$, when it has an integral structure, is necessarily a “thin group” in the sense of Sarnak [59].
Let us note that the proofs in the present paper offer an alternate route to the results from [30] and [14], where thinness is established using ping-pong and an explicit construction of cones.
These explicit cones have, nonetheless, other applications to a more detailed understanding of the monodromy groups, and which will be explored in a separate paper [31].

Note also that if the parameters $\mu,\nu$ appearing in <ref>, <ref>, and <ref> are taken to be rational, then the entries of the monodromy groups will be algebraic integers.
We can therefore apply Galois conjugation to the entries and in fact the Galois group of $\bQ$ will act via its cyclotomic quotient.
The new group will be the monodromy of another hypergeometric equation, where the parameters are obtained by multiplying the original ones by a specific integer and reducing modulo $1$.
The two monodromy groups will be abstractly isomorphic, via the isomorphism given by Galois conjugation, and since the first one was thin, so will the second one.
Note that thinness of the first group follows also because it is commensurable to a free or a surface group, a property preserved under abstract isomorphism.

To put into a broader perspective the hypergeometric examples analyzed in <ref>, let us note that Beukers–Heckman [9] were concerned with the classification of finite monodromy groups, i.e. whose image is compact and integral.
This situation can be regarded as the “rank $0$” case, where rank refers to the real rank of a real semisimple Lie group.
Fuchs, Meiri, and Sarnak [36] classified the monodromy groups acting on hyperbolic space, i.e. a “rank $1$” situation.
The results of this paper can then be placed in a “rank $2$” situation, which as we show leads to interesting geometric structures and brings in the period map and the variation of Hodge structure (which do not seem to play a direct role in the previously mentioned works).

Because of the rich geometric structures that result from assumption A (and its variant “assumption B” described below) it follows that any VHS satisfying one of these assumptions is necessarily thin.
It would be interesting to understand if there are conditions under which the converse holds.

Figure: Equivariant boundary curves in the Dwork family, on the left for the mirror quintic (rank $4$ monodromy in $\Sp_4(\bR)$ satisfying assumption A), on the right for the next case (rank $5$ monodromy in $\SO_{2,3}(\bR)$ satisfying assumption B, i.e. maximal).

§.§.§ Maximal representations

In <ref> we consider a situation analogous to “assumption A”, but which requires a different second fundamental form to be an isomorphism.
The condition on the monodromy then translates to a well-studied notion – maximal representations of surface groups [10].
Among the symplectic rank $4$ hypergeometric equations it is not possible to have a maximal representation, because of the rank $1$ unipotent at the singular point $1\in \bP^1(\bC)$.
On the other hand, it is possible to have rank $5$ hypergeometric maximal representations into $\SO_{2,3}(\bR)$.
They are classified in <ref>.

Note that a VHS which is both maximal and satisfies assumption A must necessarily be a tensor construction on the uniformizing (Fuchsian) VHS of the base Riemann surface.
A detailed investigation of the relation between maximal representations, Hodge theory, as well as a classification in higher rank of maximal hypergeometric monodromy will appear in a separate text.

Many of the properties of maximal representations are fundamentally different from the Anosov representations that satisfy assumption A.
Perhaps the most striking is the difference in their limit sets, illustrated in the figure with a numerical simulation.
The limit set of a maximal representation is always Lipschitz, whereas in the case of hypergeometric equations satisfying assumption A it is not rectifiable.
Computing its Hausdorff dimension would be an interesting challenge.

Let us finally note that the case of maximal representations also leads to a formula for Lyapunov exponents, this time expressing the top one:

\[
\lambda_1(\bV)=\frac{\deg \cV^{0,3}_{ext}}{\chi(X)}
\]

See <ref> for a discussion.

$\alpha_{\bullet}$                                         $\beta_{\bullet}$

$\left(\mu,\tfrac 12,\tfrac 12, \tfrac 12, 1-\mu \right)$

or       or

$\left(\frac{N-k_N}{2N},\frac{N-1}{2N}, \frac{1}{2},
\frac{N+1}{2N},\frac{N+k_N}{2N}\right)$

$\left(0, \frac{k_M}{M}, \frac{k_M+1}{M},\frac{M-(k_M+1)}{M},\frac{M-k_M}{M}\right)$

Any set choice from the first column is compatible with any choice from the second, subject to the condition $\alpha_{min}>\beta_{med}$, where $\alpha_{min}:=\mu$ or $\tfrac{N-k_N}{2N}$, and $\beta_{med}:=\frac{M}{2M+1}$ or $\frac{k_M+1}{M}$ depending on the choices.
The parameter $\mu$ is real, while $M,N,k_M,k_N$ are positive integers with $1<k_N<N$ and $2(k_M+1)<M$.

§.§.§ MUM Lagrangian is good, explicitly

The proof of the main conjectures from [26] is accomplished in <ref>.
In <ref>, we explain why <ref> already suffices to obtain both the formula for the sum of Lyapunov exponents, and the assertion that the “MUM Lagrangian” (Conjecture 6.4 of loc.cit.) is “good”.

In particular, this establishes <cit.> on the explicit power series that vanishes nowhere on the unit disk.

§.§.§ Formula for Lyapunov exponents, and extra punctures

The formula for the sum of Lyapunov exponents from <ref> holds in slightly greater generality.
Namely, suppose that $Y=X\setminus S$, where $S$ is a finite set of points, and $X$ is as before, i.e. finite volume hyperbolic orbifold, with $\bV\to X$ the local system underlying a VHS that satisfies assumption A.
Then the formula for the sum of Lyapunov exponents holds on $Y$ as well, even though the monodromy representation of the fundamental group of $Y$ is not log-Anosov.
This holds because the “bad locus” is empty on $Y$, if it is empty on $X$.

Let us note that this observation can be used to obtain information on classical dynamical systems, whose statement does not involve any Hodge theory, but whose proof needs it.
Namely, with the setup as above, let $\bF\to X$ be the local system giving the Fuchsian uniformization of $X$, i.e. the monodromy, valued in $\SL_2(\bR)$, is the discrete subgroup which yields $X$ after quotienting $\bH^2$.
Then the positive Lyapunov exponent is $1$ (with appropriate normalizations of the geodesic flow), and the system is uniformly hyperbolic.
For the geodesic flow on $Y$, the cocycle induced by $\bF\vert_Y$ is no longer uniformly hyperbolic.
However, its positive Lyapunov exponent is $\frac{\chi(Y)}{\chi(X)}$, where $\chi$ is the Euler characteristic.

§.§.§ Sturm's theorem on zeros and disconjugacy

It is a classical result of Sturm's in the theory of ODEs that given the equation $y''+q\cdot y=0$, between any two zeros of one solution, any other solution must vanish at least once.
Equations whose solutions have at most one zero are called disconjugate (compare with <ref>).
A geometric way to interpret Sturm's theorem is to view the differential operator as a map from the base space to $\bP^1$: the $\bP^1$ is the projectivization of the $2$-dimensional space of solutions and a point maps to the line of solutions vanishing at that point.
The discussion naturally complexifies, leading to complex projective structures on Riemann surfaces and univalence criteria, of which some of the most general are Epstein's [28].
Our uniformization results, and associated developing maps, can be viewed as saying that the corresponding Picard–Fuchs equation is “disconjugate on the universal cover”, a global restriction on its behavior.
Note that this is a fragile property, as illustrated by the conditions that arise when classifying hypergeometric equations having this property.

§.§.§ Some speculations

Our constructions produced domains of discontinuity in real and complex flag manifolds.
After reviewing the analogous situation for compact Riemann surfaces with a real structure in <ref>, we do a similar analysis for domains of discontinuity in $\LGr(V_\bC)$ and $\LGr(V_\bR)$.
Some further questions about the relationship between these domains of discontinuity are formulated in <ref>.
Some related constructions, for domains of discontinuity in complex flag manifolds, have been considered by Dumas–Sanders [25].

§.§.§ Further remarks on Lyapunov exponents

In the case when the monodromy is a lattice, or more generally has full limit set in the corresponding flag manifold, the inequality between Lyapunov exponents and degrees of Hodge bundles proved in [26] can be strict.
See for instance the work of Daniel and Deroin [23] where results along these lines are proved, albeit under the assumption that the base of the VHS is compact.

The numerical experiments of Kontsevich that led to the discovery of the formula in <ref> were further refined by Fougeron [38].
This lead to the numerical discovery of another infinite family of examples, which was proved to be thin in [30].
Note that this family, as well as the original 7 thin cases, are contained in <ref>.

For some further results on how the Lyapunov exponents depend on the representation, from the point of view of Higgs bundles, see the work of Costantini [20].

§.§.§ Outline of the text

<ref> establishes the main analytic results regarding the class of variations of Hodge structure that we consider.
The conjecture from [26] on the sum of Lyapunov exponents is established there, and the reader interested only in this formula can stop reading at <ref>, halfway through the section.
After the basic analytic properties are established, we obtain some further quantitative estimates and construct the developing maps which play a role in subsequent constructions.

<ref> classifies the hypergeometric differential equations that satisfy our assumption.
Additionally, we develop the Schwarz reflection structure of the monodromy and VHS, and explicitly analyze the period map on the real line.

<ref> contains the main Lie-theoretic results.
It is independent of the previous sections.
After recalling some of the classical notions, we introduce poset-dominated cocycles and log-Anosov representations.
The analogy with GIT is developed in <ref>, which in particular gives a criterion for proper discontinuity.
We then establish <ref> which controls the rational points on the limit curve, followed by <ref> which yields minimality of the action on the limit set.

<ref> brings together the results from the previous sections on Anosov representations and Hodge theory.
In particular we show that the domains of discontinuity are indeed uniformized by the Hodge structures, and produce some further interesting domains.
We also classify in this section the maximal hypergeometric local systems of rank $5$, producing another class of examples where a formula for the sum of Lyapunov exponents holds.

Finally, <ref> shows that using adapted metrics in the cusp, the behavior of dominated cocycles is as good as in the compact part.
We recall some further notions, including the monodromy weight filtration, that illustrate the usefulness of Hodge-theoretic techniques in the context of Anosov representations.

§.§.§ Notes and references

For more on Sturm's theorem, see [3], where a symplectic extension for real and Hamiltonian ODEs is proved.
A substantial difference with our case is that, besides complexification, our complex developing map to the Lagrangian gives a complex null-curve, whereas Arnold considers curves which are timelike.

<ref> is inspired by an analogous one, in the setting of triangle reflection groups, due to McMullen <cit.>.
An introduction to some of the applications of Lyapunov exponents and Hodge theory to dynamics is in the lecture notes of Forni and Matheus [35].

Collier, Tholozan, and Toulisse [21] consider in the broader context of Higgs bundles the case of maximal representations and their associated uniformization (what could be viewed as a counterpart “assumption B” to our “assumption A”).
It would be interesting to find, besides the hypergeometric examples in <ref>, monodromy representations that are maximal and have an integral structure.

The original observation that for weight $1$ variations of Hodge structure, one can obtain a formula for the sum of Lyapunov exponents, goes back to Kontsevich [51] and was substantially developed in the context of dynamics by Forni [37].
An extension to families of K3 surfaces can be found in [34].

More background on Anosov representations is available in Labourie's original paper [53], as well as Guichard–Wienhard [44], or the survey of Kapovich–Leeb–Porti [48], and references therein.

A Torelli theorem for generic quintic threefolds was established by Voisin [66].
However, note that Torelli theorems can fail for $1$-parameter families, for example in examples constructed by Szendröi [64].

§.§.§ Conventions

The topological notions closed, open, etc. are always for the analytic topology, unless the modifier `Zariski-' is applied.
The constants $C,\ve,$ etc. that appear in statements are distinct from one result to the next and we do not make the dependence explicit, as only the existence of such constants is sufficient.
The notation $X=O(Y)$ means that there exists $c>0$ such that $|X|\leq c Y$.
Additionally, the inequality $X\leqapprox Y$ means there exists $c>0$ such that $X\leq cY$.
Unless further specified, the constants in such inequalities are allowed to depend on fixed background geometric data, including the Riemann surface and variation of Hodge structure.

§.§.§ Acknowledgments

Most of the work on this paper was completed during the author's stay at the Institute for Advanced Study in Princeton during the academic year 2018-2019.
I am deeply grateful for the excellent working conditions.

I am grateful to Fanny Kassel for discussions around the criterion of proper discontinuity in <ref>, to Carlos Matheus for bringing my attention to the Bochi–Potrie–Sambarino paper [13], to Brian Collier for questions around the constructions in <ref>, and to Giovanni Forni for questions and suggestions that improved the presentation.

This material is based upon work supported by the US National Science Foundation under Grants No. DMS-2005470 (SF), No. DMS-1638352 (IAS), as well as
DMS-1107452, 1107263, 1107367 “RNMS: Geometric Structures and Representation Varieties” (the GEAR Network).

This research was partially conducted during the period the author served as a Clay Research Fellow.

§ HODGE THEORY AND GROWTH

In this section we use tools from Hodge theory establish the necessary growth estimates of flat sections.
These estimates imply the conjectured formula for the sum of Lyapunov exponents from [26], and will be used in <ref> to establish the Anosov property for the particular class of variations of Hodge structure under consideration.

In <ref> we set up preliminaries from Hodge theory, introduce the class of examples we study and analyze their degenerations.
The basic concept is <ref> that we call “assumption A”.
Next, in <ref> we establish the key analytic estimates that yield all subsequent results.
The basic technique is based on showing that specific functions associated to flat sections can only have local minima, and then using more quantitative information to ensure the correct growth at infinity.
Techniques from the calculus of variations provide a convenient framework.

Next, in <ref> we introduce the necessary constructions from linear algebra that will be used to produce domains of discontinuity.
We also discuss the correspondence between symplectic geometry in dimension $4$ and indefinite-orthogonal geometry in dimension $5$, arising from the isogeny of Lie groups $\Sp_4(\bR)\to \SO_{2,3}(\bR)$.
At different steps in the argument, it is more convenient to use one of the two geometries.

Finally, in <ref> we construct the uniformization maps that arise from the variations of Hodge structure satisfying assumption A.

§.§ Hodge Theory background

An introduction to variations of Hodge structures and period domains can be found in the survey of Griffiths and Schmid [43]
as well as the more recent monograph [19].
Everything that we need is concisely covered in the lectures of Griffiths <cit.>.

§.§.§ Setup

Let $\ov{X}$ be a compact Riemann surface, $D\subset X$ a possibly empty finite set of points, and let $X:=\ov{X}\setminus D$ be the complement.
Assume that $X$ has negative Euler characteristic and so has a unique complete constant curvature $-1$ metric, which we will refer to as the hyperbolic metric.
Fix also a basepoint $x_0\in X$, which we will frequently identify with its lift to the associated universal cover $\wtilde{X}$.

§.§.§ Variations of Hodge structure

Recall that a real, polarized, weight $n$ variation of Hodge structure (VHS) over $X$ consists of:

* A local system of real vector spaces $\bV\to X$, as well as a nondegenerate flat bilinear pairing on the fibers, denoted $\ip{-,-}_{I}$, which is $(-1)^n$-symmetric.
The flat connection on $\bV$ is denoted $\nabla^{GM}$ and called the Gauss–Manin connection.

* A decomposition of the complexified local system

\[
\bV_{\bC}:= \oplus \cV^{p,q} \text{ into complex subbundles}
\]

such that under complex conjugation $\ov{\cV^{p,q}}=\cV^{q,p}$.
Assume also that the decomposition is orthogonal for the indefinite hermitian pairing $\ip{v,v}_{IH}:=(\sqrt{-1})^{n}\ip{v,\ov{v}}_{I}$ and this hermitian pairing is positive, resp. negative, definite on $\cV^{p,q}$ according to whether $p$ is odd, resp. even.
Denote by $\ip{-,-}_{H}$ the positive-definite hermitian metric, called the Hodge metric, obtained from $\ip{-,-}_{IH}$ by flipping appropriately the signs on $\cV^{p,q}$.

* The Gauss–Manin connection decomposes as

\begin{align}
\label{eqn:Gauss_Manin_Higgs_field}
\nabla^{GM} = \nabla^{Ch} + \sigma + \sigma^{\dag}
\end{align}

where $\nabla^{Ch}$ is the Chern connection for a holomorphic structure on $\cV^{p,q}$, and $\sigma = \sum \sigma_{p,q}$ with

\[
\sigma_{p,q}\colon \cV^{p,q}\to \cV^{p-1,q+1}\otimes \cK_{X}
\]

holomorphic maps, called second fundamental forms.
Here $\cK_{X}$ denotes the canonical bundle of $X$, and adjoints are for the Hodge metric.

The Hodge filtration bundles are $F^p\cV:= \oplus_{p'\geq p} \cV^{p',n-p'}$ and yield holomorphic subbundles of $\bV_{\bC}$.
We will work mostly on the universal cover $\wtilde{X}$, on which the local system can be identified, using the Gauss–Manin connection, with $\wtilde{X}\times V_{\bR}$, where $V_{\bR}:=\bV(x_0)$ is the fiber over $x_0$.

Given a flat section of $\bV_{\bC}$ on $\wtilde{X}$, we can identify it with a vector $v$ in the fixed vector space.
Its $(p,q)$-components are defined by $v= \sum v^{p,q}(x)$ with $v^{p,q}(x)\in \cV^{p,q}(x)$, and satisfy, using $\autoref{eqn:Gauss_Manin_Higgs_field}$ and $\nabla^{GM}v=0$:

\begin{align}
\label{eqn:nabla_Ch_flat_components}
\nabla^{Ch}v^{p,q} =-\sigma_{p+1,q-1}v^{p+1,q-1} - \sigma^{\dag}_{p,q}v^{p-1,q+1}.
\end{align}

Note in particular that for the extremal components, i.e. $v^{n,0}$ and $v^{0,n}$, one of the two terms vanishes.

§.§.§ The weight $3$ notation

The situation of interest for this paper comes from a weight $3$ variation

\[
\bV_{\bC}:=\cV^{3,0}\oplus \cV^{2,1} \oplus \cV^{1,2} \oplus \cV^{0,3}
\]

which is equipped with a symplectic form.
We will take $\dim \cV^{p,q}=1$ for all $p,q$.

§.§.§ The weight $4$ notation

The second exterior power of the earlier weight $3$ VHS yields a weight $6$ VHS, which however splits off a $1$-dimensional piece given by the invariant symplectic form – this yields a $1$-dimensional $(3,3)$-piece which is flat and orthogonal to the rest of the VHS.
Denote by $\bW$ the remaining $5$-dimensional piece, and shift the weights by $(1,1)$ to simplify notation:

\[
\bW_{\bC} = \cW^{4,0} \oplus \cW^{3,1} \oplus \cW^{2,2} \oplus \cW^{1,3} \oplus \cW^{0,4}
\]

where the signature of the indefinite hermitian metric is $(-,+,-,+,-)$ on the respective summands, each of which is $1$-dimensional.

It is immediate to check that for instance
$\cW^{4,0}\isom \cV^{3,0}\otimes \cV^{2,1}$
$\cW^{3,1}\isom \cV^{3,0}\otimes \cV^{1,2}$
and so $\sigma_{4,0}=\id\otimes \sigma_{2,1}$.

More pictorially, we can describe the maps and bundles as:

\begin{equation}
\label{eqn:W_hodge_bundles_sigma_maps}
\begin{tikzcd}
\cW^{4,0}
\arrow[r, bend right, swap, "\sigma_{4,0}"]
\arrow[r, bend left, leftarrow, "{\sigma_{4,0}^\dag}"]
\cW^{3,1}
\arrow[r, bend right, swap, "\sigma_{3,1}"]
\arrow[r, bend left, leftarrow, "{\sigma_{3,1}^\dag}"]
\cW^{2,2}
\arrow[r, bend right, swap, "\sigma_{3,1}^t"]
\arrow[r, bend left, leftarrow, "\ov{\sigma_{3,1}}"]
\cW^{1,3}
\arrow[r, bend right, swap, "\sigma_{4,0}^t"]
\arrow[r, bend left, leftarrow, "\ov{\sigma_{4,0}}"]
\cW^{0,4}
\end{tikzcd}
\end{equation}

where we use the identification of duals to write the right part of the diagram of second fundamental forms.

§.§.§ Deligne extension

As explained in <ref> below, orbifold points are treated by passing to the appropriate finite cover.
Suppose now that $d\in D=\ov{X}\setminus X$ is a cusp, and further that the monodromy of the local system around $d$ is unipotent.
Then there exists a canonical choice of extension $\bV_{ext}$ of the local system in a neighborhood of $d$, called the Deligne extension.
Additionally, the Hodge bundles $\cV^{p,q}$ have extensions $\cV^{p,q}_{ext}$ as well, obtained by first extending the Hodge filtration bundles.

A weight $3$, rank $4$, variation of Hodge structure will be said to satisfy assumption A if

\[
\sigma_{2,1}\colon \cV^{2,1}_{ext}\to \cV^{1,2}_{ext}\otimes \cK_{\ov{X}}(\log D)
\]

is an isomorphism, where $\cK_{\ov{X}}(\log D)$ denotes the bundle of meromorphic $1$-forms on $\ov{X}$ with logarithmic (i.e. $\tfrac{dz}{z}$) poles at the points in $D$.

Analogously, a weight $4$, rank $5$ variation of Hodge structure will be said to satisfy assumption A if

\[
\sigma_{1,3}\colon \cW^{1,3}_{ext} \to \cW^{0,4}_{ext}\otimes \cK_{\ov{X}}(\log D)
\]

is an isomorphism.

Note that a weight $3$ VHS satisfies assumption A if and only if its associated weight $4$ VHS via the construction in <ref> satisfies it.
Indeed, we have that $\sigma_{1,3}=\sigma_{4,0}^{t}$ and the earlier remarks about the relation between second fundamental forms give the equivalence.

From now on, we assume that our variations of Hodge structure satisfy assumption A.

§.§.§ Notation for norms

Several norms will appear throughout the arguments below.
The flat indefinite hermitian form is denoted $\ip{-,-}_{IH}$, and $\norm{v}^2_{IH}:=\ip{v,v}_{IH}$.
Similarly $\ip{-,-}_H$ and $\norm{-}^2_{H}$ will denote the positive-definite Hodge pairing and norm.
For tangent vectors or covectors, we will use the norm coming from the fixed background hyperbolic metric on $X$, and which will be denoted $\norm{-}_{hyp}$.
Note that the hyperbolic metric can be viewed as a singular hermitian metric on (the dual of) $\cK_{X}(\log D)$.

When it will be clear from the context (i.e. which bundle the vector $v$ belongs to), the notation $\norm{v}$ will denote either the Hodge norm, or the hyperbolic norm.
When speaking of “absolute” constants, or uniformly bounded quantities, we will refer to constants that only depend on the original VHS over $X$, and the fixed hyperbolic metric on $X$.
The notation $A\geqapprox B$ means that there exists an absolute constant $c>0$ such that $c\cdot A\geq B$.

If the weight $3$ VHS satisfies assumption A, then there exists a constant $C>0$ such that

\[
C\geq	\norm{\sigma_{2,1}} \geq \frac{1}{C}
\]

where we view $\sigma$ as a section of $\left(\cV^{2,1}\right)^{\dual}\otimes \cV^{1,2}\otimes \cK_{X}(\log D)$ and each term of the tensor product is equipped with its Hodge, resp. hyperbolic metric.
The same holds for $\norm{\sigma_{1,3}}$ in the case of a weight $4$ VHS.

Furthermore, in weight $3$, at any of the punctures, the Hodge numbers of the limiting mixed Hodge structure, and corresponding monodromy, are the second and fourth in <ref>.

Conversely, if only the two listed situations occur at the punctures, and $\sigma_{2,1}$ does not vanish on $X$, then assumption A is satisfied.

The possible Hodge numbers of a degeneration.
From left to right: first, the original (pre-degeneration) numbers,
then a rank $1$ unipotent with an invariant line,
then a rank $1$ unipotent with an invariant Lagrangian,
then a maximally unipotent monodromy.

Recall that on the punctured unit disc with coordinate $q$, the hyperbolic metric is (up to a constant factor) equal to $\tfrac{|dq|^2}{\left(|q|\log|q|\right)^2}$.
This is the metric $\frac{|d\tau|^2}{(\Im \tau)^2}$ descended from the upper half-plane by $\tau \mapsto \exp(\twopii \tau)=:q$.
In particular, the vector field $q\partial_q$ has norm $\frac{1}{|\log |q||}$ and thus the logarithmic differential $\frac{dq}{q}$ has unbounded norm.

Let us see that in the standard weight $1$ degeneration the bound claimed in <ref> holds.
Specifically, we have that $\bC^2 = v^{1,0}(\tau) \oplus \ov{v^{1,0}(\tau)}$ where $v^{1,0}(\tau)=a+\tau b$.
Then $\nabla_{\partial_{\tau}}v^{1,0} = b$, and so as sections of line bundles, we have that $\sigma_{1,0}=\frac{b}{v^{1,0}\cdot \partial_{\tau}}$.
Computing the Hodge and hyperbolic norms yields $\norm{v^{1,0}}^{2}=2\Im \tau$, $\norm{b}^{2}=\frac{2}{\Im\tau}$ and $\norm{\partial_{\tau}}^2=\frac{1}{(\Im \tau)^2}$ from which the bounded (in fact, constant) norm of $\sigma^{1,0}$ follows.

It is immediate that all questions are local at the punctures, since on a compact set a continuous nonvanishing function is bounded above and below.
In any of the cases, it suffices to establish the bounds for the nilpotent orbit corresponding to the degeneration, and in fact the $\SL_2$-orbit as well, since by the results of Schmid <cit.>, the metrics in the two cases are comparable by uniform constants, which in fact go to $1$ as $\Im \tau\to +\infty$.

In the maximally unipotent monodromy case, the behavior of the nilpotent orbit is the same as that of the third symmetric power of the example analyzed in <ref>, so the conclusions follow.

To proceed with the next cases, we will freely use standard terminology in the subject, see e.g. <cit.> and references therein.
Fix a symplectic basis $e_1,f_1,e_2,f_2$ of $V_\bR$ with $\ip{e_i,f_i}_I=1$ and $0$ otherwise.
Let $N:=\log T$ be the logarithm of the monodromy, working under the assumption that $T$ is purely unipotent.
The elliptic part, when present, does not affect the metric estimates (see also <ref> below).
Recall also that the nilpotent orbit on a (subset of) the upper half plane is given by the formula $e^{\tau N}F^{\bullet}_{lim}$ where $F^{\bullet}_{lim}$ is the limit of the Hodge filtration (in the Deligne extension of the local system).
For asymptotic questions, we can also work under the assumption that the limiting mixed Hodge structure is $\bR$-split, so it suffices to describe the spaces $I^{p,q}$, subject to the polarization and complex-conjugation axioms of a polarized mixed Hodge structure.
Then the limiting Hodge filtrations are given by

\[
F^{p}_{lim} = \bigoplus_{r\geq p, \forall s}I^{r,s}.
\]

The second diagram in <ref> corresponds to $Ne_1=-f_1$ and $0$ otherwise (the reason for $-f_1$ instead of $f_1$ is to have the correct alternating signature below).
Then the $\bR$-splitting is given by

\begin{align}
\label{eqn:nilpotent_orbit_rank1_line}
\begin{split}
I^{3,0} & = e_2 + \sqrt{-1}f_2 \\
I^{0,3} & = \ov{I^{3,0}}
\end{split}
\begin{split}
I^{2,2} & = e_1\\
I^{1,1} & = f_1
\end{split}
\end{align}

so that for the Hodge filtrations and nilpotent orbit we find:

\begin{align*}
\begin{split}
F^3_{lim} & = span(e_2 + \sqrt{-1}f_2)\\
F^2_{lim} & = F^3_{lim} + e_1\\
F^1_{lim} & = F^2_{lim} + f_1
\end{split}
\begin{split}
F^3(\tau) & = F^3_{lim}\\
F^2(\tau) & = F^3(\tau) + (e_1 - \tau f_1)\\
F^1(\tau) & = F^2(\tau) + f_1
\end{split}
\end{align*}

The desired estimate on $\sigma^{2,1}$ now follows exactly as in <ref>.

The third diagram of Hodge numbers corresponds to the nilpotent operator acting by $Ne_i=f_i$ and $I^{p,q}$-components:

\begin{align}
\label{eqn:nilpotent_orbit_rank1_Lagrangian}
\begin{split}
I^{3,1} = e_1 + \sqrt{-1}e_2\\
I^{1,3} = e_1 - \sqrt{-1}e_2\\
\end{split}
\begin{split}
I^{2,0} = f_1 + \sqrt{-1}f_2\\
I^{0,2} = f_1 - \sqrt{-1}f_2\\
\end{split}
\end{align}

which leads to the limit Hodge filtrations and nilpotent orbit:

\begin{align*}
\begin{split}
F^3_{lim} & = e_1 + \sqrt{-1}e_2\\
F^2_{lim} & = F^3_{lim} + (f_1+\sqrt{-1}f_2)\\
F^1_{lim} & = F^2_{lim} + (e_1+\sqrt{-1}e_2)
\end{split}
\begin{split}
F^3(\tau) & = (e_1+\tau f_1)+\sqrt{-1}(e_2+\tau f_2)\\
F^2(\tau) & = F^2_{lim}\\
F^1(\tau) & = F^2_{lim} + \tau(f_1+\sqrt{-1}f_2)
\end{split}
\end{align*}

Visibly, the second fundamental form $\sigma_{2,1}$ vanishes.

§.§.§ Orbifold points, elliptic parts

The arguments below, and definitions above, apply also to the situation where $X$ is an orbifold, under the following assumption:
There exists a finite orbifold cover $X'\to X$ such that $X'$ is a Riemann surface and assumption A applies to the VHS pulled back to $X'$.
In other words, there exists a torsion-free finite index subgroup $\pi_1(X')$ of $\pi_1^{orb}(X)$ such that the assumptions and definitions apply to $X'$.

Additionally, it is possible for the monodromy matrix $T$ to decompose into commuting elliptic and unipotent parts $T=T^{u}\cdot T^e$.
Furthermore, when the monodromy is not integral it is possible for $T^e$ to be of infinite order.
Nonetheless, the metric estimates of <ref> apply and this also leads to cases that satisfy assumption A in the hypergeometric examples.

§.§ Gradient and index estimates

We proceed with a weight $4$ VHS as in <ref>.
In this section, we establish the basic gradient estimates and show that certain functions can only have critical points which are local minima, and in fact at most one local local minimum can occur.

The conjectured formula for the sum of Lyapunov exponents from [26] follows from <ref>, and also the slightly stronger statement that the MUM Lagrangian is good, see <ref>.

§.§.§ Setup

Let $w\in W_\bR\setminus 0$ be a null, or a positive vector.
In other words $\ip{w,w}_{I}\geq 0$, and if $w$ is a positive vector we normalize it to $\ip{w,w}_I=1$.
Using that $w$ is real, for any Hodge decomposition

\[
w = w^{4,0} \oplus w^{3,1}\oplus w^{2,2} \oplus w^{1,3} \oplus w^{0,4}
\]

we thus have

\[
2\norm{w^{1,3}}^{2} - \left(2\norm{{w}^{0,4}}^2 + \norm{w^{2,2}}^2\right) = \ip{w,w}_I \geq 0
\]

where the last inequality holds because the vector is nonzero.
In particular, note that

\begin{align}
\label{eqn:basic_non_degeneracy}
\begin{split}
w^{1,3}\neq 0 & \text{ and } \norm{w^{1,3}} \geq \norm{w^{0,4}}\\
\norm{w^{1,3}} \geq 1 & \text{ if }\ip{w,w}_{I}=1
\end{split}
\end{align}

Introduce now on $\wtilde{X}$ the function

\[
f_w(x) := \norm{w^{0,4}(x)}^2
\]

relative to the Hodge decomposition at $x\in \wtilde{X}$.
When $w$ is fixed and clear from the context, we will omit it as a subscript.

§.§.§ The directing vector field

Recall also that we have the second fundamental form $\sigma_{1,3}\colon \cW^{1,3}\to \cW^{0,4}\otimes \cK_X$.
Since $\sigma_{1,3}$ is nowhere vanishing by <ref> and $w^{1,3}$ doesn't vanish on $\wtilde{X}$ either, there exists a unique vector field $V_w$ such that

\[
\sigma_{1,3}(V_w)w^{1,3} = -w^{0,4} \text{ or in local coordinates }
V_w = \frac{-w^{0,4}}{\sigma_{1,3}\left(w^{1,3}\right)}.
\]

Crucially, note that for the definition of $V_w$, both the nonvanishing of $\sigma_{1,3}$ and $w^{1,3}$ were used.

Let $V=V_w$ and $f=f_w$ be as above.
Equip $\wtilde{X}$ with the hyperbolic metric.

* For the Lie derivative $\cL_V$ we have that

\[
\cL_V f = 2f \text{ and }V, \grad f \text{ are proportional.}
\]

Furthermore we have

\begin{align}
\label{eqn:gradf_V_2f_identity}
\norm{\grad f}_{hyp}\cdot \norm{V}_{hyp} = 2f.
\end{align}

* We have the following “steepness” estimate

\begin{align}
\label{eqn:steepness_basic}
\norm{V}_{hyp}\leqapprox 1 \text{ and thus } \norm{\grad f}_{hyp}\geqapprox f.
\end{align}

* If furthermore $\ip{w,w}_I=1$ then we also have the inequality

\begin{align}
\label{eqn:steepness_low_f}
\norm{V}_{hyp}\leqapprox f^{1/2} \text{ and so }
\norm{\grad f}_{hyp} \geqapprox f^{1/2}.
\end{align}

Note that the inequality in the last point is stronger than the one in the second point only when $f$ is sufficiently small.

Let us compute the differential of $f$:

\begin{align*}
df  & = d \ip{w^{0,4},w^{0,4}}_{H}\\
& = \ip{\nabla^{Ch}w^{0,4},w^{0,4}}_H +
\ip{w^{0,4},\nabla^{Ch}w^{0,4}}_H\\
& = \ip{-\sigma_{1,3}w^{1,3},w^{0,4}}_H +
\ip{w^{0,4},-\sigma_{1,3}w^{1,3}}_H
\end{align*}

Note that the two terms are complex-conjugates of one another.
Using the defining property of $V$ that $\sigma_{1,3}(V)w^{1,3}=-w^{0,4}$ it follows that $\cL_V f = 2f$ as claimed.
Note that for a complex $1$-form $\alpha$ and vector field $V$ we have that $\iota_V\ov{\alpha}=\ov{\iota_V \alpha}$ for the insertion of a vector field.

Next, observe that $\sqrt{-1} V$ satisfies $\cL_{\sqrt{-1}V}f=0$, since the first term in the formula for $df$ would yield $\sqrt{-1}f$ and the second one its complex-conjugate, so the two terms cancel out.
It follows that $V$ and $\grad f$ are proportional.
Combining the two formulas (all vectors measured in the hyperbolic metric)

\[
\cL_V f = 2f \quad \text{ and }\cL_{\grad f} f = \norm{\grad f}^{2}
\]

it follows that $\norm{\grad f}^2 V = 2f \grad f$, or by taking norms we find that

\[
\norm{\grad f}\cdot \norm{V} = 2f.
\]

Note that if we rescale the background hyperbolic metric by a factor of $e^{2\lambda}$, so that $\norm{V}$ scales by $e^{\lambda}$, then $\grad f$ scales by $e^{-2\lambda}$ and thus $\norm{\grad f}$ will scale by $e^{-\lambda}$ so the stated identity is scale-invariant.

To establish the estimate $1\geqapprox \norm{V}$, note that as a section of the tangent bundle $TX$, we have that $V=w^{0,4}/\sigma_{1,3}w^{1,3}$ by definition.
Now from <ref> it follows that $1\leqapprox \norm{\sigma_{1,3}}\leqapprox 1$.
From the condition $\ip{w,w}_I\geq 0$, see <ref>, we also have that $\norm{w^{0,4}}\leq \norm{w^{1,3}}$ and so the claimed estimate follows.

To establish also the estimate $\norm{V}\leqapprox f^{1/2}$, observe that we can also use $\norm{w^{1,3}}\geq 1$, as well as $\norm{w^{0,4}}=f^{1/2}$, which can be plugged into the definition of $V$.

The estimates we have obtained can now be used to analyze the Morse index of $f$, as well as establish some growth properties.

Let $f=f_w$ be as above.

* The only critical points of $f$ are local minima, which occur when $w^{0,4}(x)=0$.
Furthermore these are nondegenerate local minima.

* If $\ip{w,w}_I\geq 0$ then $f$ has at most one critical point, and furthermore $\inf_{\wtilde{X}}f=0$.

* If $\ip{w,w}_I > 0$ then $f$ has precisely one critical point.

It follows from <ref> that if $\grad f$ vanishes, then necessarily $f$ also vanishes.
So the only critical points are local minima.
To check that they are nondegenerate, observe that $w^{0,4}$ is a holomorphic section of $\cW^{0,4}$, so it suffices to establish that it only has simple zeros.
But $\nabla^{Ch}w^{0,4}=\sigma_{1,3}w^{1,3}\neq 0$ pointwise, so $w^{0,4}$ cannot vanish to second order or higher at a point.

To establish (ii), namely that $f$ cannot have two critical points (necessarily local minima), we would like to invoke a mountain pass lemma.
Most formulations assume that $f$ already satisfies a Palais–Smale condition, which is not (yet) available for $f$, so we will use it in the form which produces a Palais–Smale sequence.
Specifically, by <cit.>, suppose that $f(x_0)=f(x_1)=0$, with $x_0\neq x_1$, and let $r_0,\ve_0>0$ be such that $x_1\notin B(x_0,r_0)$ and $f\vert_{\partial B(x_0,r_0)}\geq \ve_0$.
Then there exists a sequence $y_n$ such that $f(y_n)\geq \ve_0$ and $\norm{\grad f(y_n)}\to 0$.
The assumptions are satisfied, by picking sufficiently small $r_0,\ve_0>0$.
But the conclusion contradicts the steepness estimate <ref>, since that estimate shows that $\norm{\grad f(y_n)}\to 0$ implies $f(y_n)\to 0$.

To check that $\inf_{\wtilde{X}}f=0$, let $c\geq 0$ be this infimum.
By the Ekeland variational principle, see e.g. <cit.>, there exists a sequence $x_n$ with $f(x_n)\to c$ and $\norm{\grad f(x_n)}\to 0$.
Again from the steepness condition <ref> it follows that $f(x_n)\to 0$, i.e. $c=0$.

To establish (iii), namely that when $\ip{w,w}>0$ (say normalized to $1$), $f$ must achieve a minimum, we will show that the gradient flow converges, in fact, to a minimum located at finite distance.
Specifically, start at some $x_0\in \wtilde{X}$ and let $\gamma\colon [0,t_0)\to \wtilde{X}$ be the unique curve with $\gamma(0)=x_0$ and $\dot{\gamma}(t)=-\grad f(\gamma(t))$, and $t_0>0$ is the maximal existence time (along the way we will establish $t_0=+\infty$).
Set $g(t):=f(\gamma(t))$, so $-\dot{g}(t)=\norm{\grad f(\gamma(t))}^2$ which by the other steepness estimate <ref> satisfies $-\dot{g}(t)\geq \ve_0\cdot g(t)$ for some $\ve_0>0$.
It follows that $\dot{g}/g \leq -\ve_0$, so by integrating this logarithmic derivative (and exponentiating) on any interval $[a,b]\subset[0,t_0)$ we have $g(b)\leq g(a)e^{-\ve_0 \cdot t}$.

The distance traversed by the curve $\gamma(t)$ on the time interval $[a,b]$ is $\int_{a}^{b}\norm{\dot{\gamma}(t)}dt$ which can be estimated by Cauchy–Schwartz:

\begin{align*}
\left(\int_{a}^{b}\norm{\dot{\gamma}(t)}dt\right)^2 & \leq
\left[\int_{a}^{b}\norm{\dot{\gamma}(t)}^2dt\right] \cdot (b-a)\\
& = \left[\int_{a}^b |\dot{g}|dt\right] (b-a)\\
& = (g(a)-g(b))\cdot (b-a) \leq g(a)\left(1-e^{-\ve_0\cdot (b-a)}\right)(b-a)
\end{align*}

Take now $b-a=1$ and use the estimate $g(a)\leq e^{-\ve_0\cdot a}g(0)$ to find that the length of the curve on a time interval $[x,x+1]$ is bounded by $\left(g(0) e^{-\ve_0\cdot x}(1-e^{-\ve_0})\right)^{1/2}$.
Any interval $[x,y]\subseteq [0,\infty+)$ can now be divided into $\lceil y-x\rceil$ consecutive intervals $[x,x+1], [x+1,x+2], \ldots \left[\lfloor y\rfloor,y\right]$, and applying the previous estimate we find that the total length is bounded by

\begin{align}
\label{eqn:distance_bound_O1}
\begin{split}
\dist(\gamma(x),\gamma(y))& \leq g(0)^{1/2}(1-e^{-\ve_0})^{1/2}\sum_{n=0}^{\infty} e^{-\ve_0\cdot (x+n)/2}\\
\leq g(0)^{1/2}\frac{\left(1-e^{-\ve_0}\right)^{1/2}}{1-e^{-\ve_0/2}}\\
& \leq c_3 \cdot g(0)^{1/2}
\end{split}
\end{align}

where $c_3$ is some fixed constant.
In particular, $\gamma$ stays in a compact set $K$, for which there exists an $\ve_1(K)>0$ such that the gradient flow exists for time $\ve_1$ at any point in $K$, therefore the gradient flow exists for all (positive) times.

The limit point of the curve $\gamma(t)$ is necessarily critical, thus a local minimum, and hence the global minimum with value $0$ as claimed.

<ref> already implies the formula for Lyapunov exponents conjectured in <cit.>.
Indeed <ref> below shows that the seven “good” families of loc. cit. satisfy assumption A (and in fact our <ref> and <ref> provide infinitely many other examples).

Next, the proof of <cit.>, in the case at hand, starts with a Lagrangian subspace $L$ of $V_\bR$ and considers the corresponding vector $w_{L}\in W_{\bR}$ obtained by wedging two independent vectors in $L$.
The argument then applies verbatim, with $u=w_L$, and observing that the function $\norm{u_z}$ in loc. cit. agrees with $f_{w_L}$ in the present text.
Next, $f_{w_L}$ has at most one zero in all of $\bH$, by <ref>(ii).
Then the inequality (15) in loc. cit. is an equality, up to adding a constant on the lhs if the hyperbolic disc $D(u)$ contains the one allowed zero.
The chain of deductions following inequality (15) in loc. cit. then becomes a chain of equalities, if we take into account that $\lim_{T\to +\infty}\frac{1}{T}C=0$ for any constant $C$.

Let us also explain why Conjecture 6.4 of [26] follows from <ref>(ii).
The stated conjecture is equivalent to the statement that for a particular choice of Lagrangian $L \subset V_{\bR}$, the locus of vanishing of the function $\norm{u_z}$ is empty.
The Lagrangian has the property that it fixed by the monodromy around a cusp of the Riemann surface.
More generally, suppose that there exists an infinite order element $\gamma\in \pi_1(X)$ which fixes a Lagrangian $L\subset V_{\bR}$.
Then if $w_{L}\in \Lambda^2L$ denotes the (necessarily isotropic) vector in $W_{\bR}$, the associated function $f_{w_L}$ of <ref> cannot have any zeros on $\wtilde{X}$.
Indeed, the function and hence the set of zeros will be invariant under $\gamma$, any orbit of $\gamma$ is infinite on $\wtilde{X}$, while the theorem just proved shows that there is at most one zero.

We next establish a quantitative growth property for the norm of positive-definite vectors.

Fix $w\in W_\bR$ satisfying $\ip{w,w}_{I}=1$, let $x_0\in \wtilde{X}$ be the unique minimum of $f:=f_w$.
There exist uniform constants $\ve,C_1,C_2>0$ (independent of $w$ or $x_0$) such that for any $x\in \wtilde{X}$ we have:

\[
f(x)\geq \frac{1}{C_1}e^{\ve \dist(x_0,x)} - C_2
\]

where the distance is in the hyperbolic metric on $\wtilde{X}$.

We will prove the equivalent

\[
\dist(x_0,x) \leq c_1' + \frac{1}{\ve}\log\left(1 + f(x)\right)
\]

As in the proof of <ref>, take $\gamma\colon [0,\infty)\to \wtilde{X}$ to be the gradient flow line satisfying $\dot{\gamma}(t)=-\grad f$.
Next, observe that if we started at a point $x'$ where $f(x')\leq 2$, then by the estimate in <ref>, we have that $\dist(x_0,x')\leq c_3$ for some absolute constant $c_3$.
So it suffices to prove that

\[
\dist(x,x')\leq c_1''+\frac{1}{\ve}\log(c_2+f(x))
\]

where $f(x)>2$ and $x'=\gamma(t')$ is such that $f(x')=\frac{f(x)}{2^k}\in [1,2)$, for some integer $k\geq 1$.

Setting as before $g(t):=f(\gamma(t))$ we have that $\dot{g}=-\norm{\grad f}^2$ and so we need to estimate $\int_{0}^{t'}|\dot{g}(t)|^{1/2}dt$.
To do so, we will use the other steepness inequality <ref> which gives

\[
-\dot{g}\geq \ve g^2 \text{ for some fixed }\ve>0.
\]

Let $t_1>0$ be such that $g(t_1)=\frac{1}{2}g(0)$, which exists since the gradient flow line eventually gets arbitrarily close to the minimum by <ref>.
Let us rewrite the differential inequality as

\[
\frac{d}{dt}\frac{1}{g(t)}\geq \ve
\]

and integrate from $0$ to $t_1$ to find that

\[
\frac{1}{g(0)}\geq \ve t_1 \quad \text{ i.e. } t_1 \leq \frac{1}{g(0)\ve}.
\]

In particular, since $g(0)\geq 2$ we have that $t_1$ is uniformly bounded.
Next, we apply Cauchy–Schwartz again to find

\begin{align*}
\left(\int_{0}^{t_1}|\dot{g}(t)|^{1/2}\right)^2 & \leq
\left(\int_0^{t_1}|\dot{g}(t)|(t+1)dt\right)
\left(\int_{0}^{t_1}\frac{dt}{t+1}\right)\\
& \leq (t_1+1)\left(\int_{0}^{t_1}|\dot{g}(t)|dt\right)
\log(1+t_1)\\
& = (t_1+1)\cdot \frac{g(0)}{2}\cdot \log(1+t_1)
\end{align*}

Using that $\log(1+x)\leq x$ for $x\geq 0$, that $t_1\leq \frac{1}{g(0)\cdot \ve}$ but is also uniformly bounded above, it follows that the distance is uniformly bounded by some absolute constant $c_4$, independently of $g(0)$.
Repeating this argument for times $t_1<t_2<\cdots <t_k$ where $k=\lfloor\log_2 g(0)\rfloor $, we find that

\[
\dist(x,x')\leq c_4 \cdot k \leq c_4 \cdot \log_2 f(x).
\]

This implies the desired bound

\[
\dist(x,x_0)\leq c_3 + c_4 \log_2(1+f(x))
\]

where the additive factor of $1$ inside the logarithm is to ensure that the bound is valid even if $f(x)\leq 1$.

The same proof yields the following variant for isotropic vectors.
Indeed, the proof above shows that in logarithmic time the gradient flow brings the value of the function to within $O(1)$, and it suffices to get the following conclusion:

Suppose that $w\in W_{\bR}\setminus \{0\}$ is an isotropic vector, such that $f_w$ achieves its (necessarily unique) minimum at $x_0$.
Then there exist constants $C_1,C_2,\ve>0$, uniform for $w$ varying in a compact set and still satisfying the assumptions, such that

\[
f_w(x)\geq \frac{1}{C_1}e^{\ve \dist(x_0,x)} - C_2 \quad \forall x\in \wtilde{X}.
\]

The above growth estimates will be used in <ref>, see e.g. <ref>, to establish that the monodromy representation is log-Anosov (after introducing the appropriate Lie-theoretic notions in <ref>).

§.§ Symplectic–Orthogonal dictionary

Before we can draw more consequences from the results of the previous section, we need to introduce some further concepts from symplectic geometry and relate it to the orthogonal group.
An exposition of many of the constructions described here is in the lectures notes [4], with the exception of the concept of “electron” which we introduce in <ref> below.

§.§.§ Setup

Let $V$ be a real $4$-dimensional symplectic vector space.
We will typically write $I(v,w)$ for the symplectic pairing of two vectors.
When a basis will be useful, we take $e_1,e_2,f_1,f_2\in V$ to be one, satisfying $I(e_i,f_i)=1$ and vanishing otherwise.

We will also need $W:=\Lambda^2_\circ V$, the subspace of $\Lambda^2V$ of vectors in the kernel of the symplectic form.
Indeed $\Lambda^2 V$ is equipped with a symmetric inner product of signature $(3,3)$, given by wedging and using the volume form on $\Lambda^4 V$.
Restricted to $W$, the inner product has signature $(2,3)$.

The dictionary between the symplectic geometry of $V$ and the Minkowski geometry of $W$ is summarized in <ref>.
Many constructions work over $\bR$ and over $\bC$ – when a specific field is considered it will appear as a subscript; otherwise, the stated construction works over both fields.

§.§.§ The corresponding groups

The map $\Sp(V)\to \SO(W)$ gives over $\bR$ the double cover $\Sp_4(\bR)\to \SO_{2,3}(\bR)$.
Restricted to maximal compact subgroups this yields $\bbU(2)\to \operatorname{S}(\Orthog_2(\bR)\times \Orthog_3(\bR))$.
When dealing with Hodge decompositions, the group $\bbU(1,1)$ will arise as the stabilizer of $F^2=V^{3,0}\oplus V^{2,1}$, and it maps to $\operatorname{S}(\Orthog_2(\bR)\times \Orthog_{1,2}(\bR))$.

§.§.§ Lagrangian Grassmannian

Let $\LGr(V)$ denote the Grassmannian of Lagrangian subspaces of $V$.
The natural embedding $\LGr(V)\into \bP(W)$ identifies the Lagrangian Grassmannian with the quadric of primitive vectors, when viewing $W$ as a subset of $\Lambda^2 V$.
Equivalently, $\LGr(V)$ is the quadric of isotropic vectors in $W$.

Over $\bR$ it can be useful to consider the Grassmannian of oriented Lagrangians, which is a double cover of $\LGr(V_{\bR})$ and will be denoted $\LGr^+(V_{\bR})$.
If $\bP^+(W_{\bR})$ is the double cover of $\bP(W_{\bR})$, then we have the natural inclusion $\LGr^{+}(V_{\bR})\subset \bP^+(W_{\bR})$.

§.§.§ Causal structure

At an isotropic line $[w]$, the tangent space to the quadric of null vectors can be identified with the linear space

\[
T_{[w]}\LGr(V) = \Hom\left([w],\leftquot{[w]}{[w]^\perp}\right)
\]

where $[w]^\perp$ is the orthogonal of $[w]$ for the indefinite pairing.
Over $\bR$ the quotient $\leftquot{[w]}{[w]^\perp}$ carries a natural inner product of signature $(1,2)$ and so $Q^0(W_\bR)$ carries a natural conformal class of such inner products.
This determines a causal structure invariant under the corresponding group.
For this reason $\LGr(V_\bR)$ is also denoted $\Ein^{1,2}(W_\bR)$ as a “conformal Einstein universe”.
It is a compactification of flat Minkowski space of signature $(1,2)$, which can be obtained by taking the complement of $[w]^{\perp}\cap\Ein^{1,2}(W_\bR)$, for any $[w]\in \Ein^{1,2}$ (the described intersection is the union of all null rays through $[w]$).

§.§.§ Photons

Let $l\subset V$ be a line.
It determines a flag $l\subset l^\perp\subset V$, where $\perp$ is taken for the symplectic form.
Define the photon associated to $l$ by

\[
\phi(l):= \left\lbrace L\subset V \colon L \text{ is Lagrangian and }l\subset L \right\rbrace \subset \LGr(V).
\]

The definition makes sense both over $\bR$ and over $\bC$, with associated photons denoted $\phi_{\bR}(l)$ (or just $\phi(l)$) and $\phi_{\bC}(l)$.

Note that any Lagrangian $L$ satisfying $l\subset L$ necessarily satisfies (by taking symplectic-orthogonal complements) $L\subset l^\perp$.
Therefore $\phi(l)$ is naturally parametrized by $\bP\left(\leftquot{l}{l^\perp}\right)$.
In the embedding $\LGr(V)\subset \bP(W)$ we have the natural identification $\phi(l)\isom \bP(l\wedge l^\perp)$.

In fact, any isotropic $2$-dimensional space $\xi_2\subset W$ determines a photon $\bP(\xi_2)\subset \bP(W)$.
For the Minkowski structure on $\LGr(V_{\bR})$ the curves $\phi(l)$ are lightlike, i.e. their tangent directions are null vectors.

Let $F^2\subset V_\bC$ be a $2$-dimensional subspace, Lagrangian for the symplectic form and with the restriction of the Hermitian pairing on $F^2$ of signature $(1,1)$.
Define the subset

\[
\beta(F^2):=\left\lbrace L\in \LGr(V_\bR)\colon L_\bC \cap F^2 \neq \left\lbrace 0 \right\rbrace \right\rbrace
\]

and call it the electron associated to $F^2$.

§.§.§ On electrons

Some of the reasons for calling $\beta(F^2)$ above an “electron” are:

* In $\LGr(V_{\bR})$ with its Minkowski structure, the curves $\beta(F^2)$ are timelike (so they correspond to a particle moving below the speed of light).
It will follow from <ref> that when the electron approaches the boundary it tends to a photon.

* In the physics literature, electrons are denoted by $e^-$ or $\beta^-$, and $\beta$ is also convenient for the terminology of “bad” Lagrangians, see <ref> below.

To geometrically identify the set $\beta(F^2)$, note that if a real Lagrangian $L$ satisfies $L_\bC\cap F^2 = \bC\alpha$ for $\alpha\in F^2$, then we have the following properties:

* $L_{\bC} = span(\alpha,\ov{\alpha})$

* $\ip{\alpha,\ov{\alpha}}_{I}=0$

The first follows because $\alpha$ and $\ov{\alpha}$ are never proportional (since $V_{\bC}=F^2\oplus \ov{F^2}$ so $F^2$ cannot contain a real vector) and because $L$ is a real subspace, so $\ov{\alpha}\in L_{\bC}$.
The second property follows because $L$ is Lagrangian.

This discussion implies that we have a natural identification

\[
\beta(F^2) = \left\lbrace [\alpha]\in \bP(F^2) \colon \ip{\alpha,\ov{\alpha}}_{I}=0 \right\rbrace \subset \bP(F^2)\isom \bP^1(\bC)
\]

In other words the photon identifies with the equator of null vectors in $\bP(F^2)$; this equator separates the upper/lower half planes of positive/negative directions.

The “moduli space” of electrons is the Grassmannian of complex Lagrangians where the hermitian pairing has signature $(1,1)$, which as a homogeneous space is:

\begin{align}
\label{eqn:space_of_electrons}
\rightquot{\Sp_4(\bR)}{U(1,1)}=L\LGr^{1,1}(V_\bC)
\end{align}

and is naturally a complex manifold.

§.§.§ Electrons via second exterior power

Just like there is a definition of the photon as $\phi(l)=\bP(l\wedge l^\perp)\subset \Ein^{1,2}\subset \bP(W)$, an analogous construction exists for electrons.
If $V_\bC=F^2\oplus \ov{F^2}$ with $F^2$ a complex Lagrangian of signature $(1,1)$, then in $W_\bC$ we have:

\[
W_{\bC} = \underbrace{\left(\Lambda^2 F^2 \oplus \ov{\Lambda^2F^2}\right)}_{N_{\bC}} \oplus N^{\perp}_{\bC}
\]

The subspace $N_\bC$ is invariant under complex conjugation, hence leads to a real subspace $N\subset W_{\bR}$.
Its orthogonal complement also comes from a real subspace $N^\perp$.

Suppose now that $L\in \beta(F^2)$ is a real Lagrangian.
Then its coordinate is orthogonal to $\Lambda^2F^2$, since a pair of subspaces intersect if and only if their coordinates wedge to zero (so, are orthogonal for the specified inner product).
Because $L$ is real it is therefore also orthogonal to $\ov{\Lambda^2F^2}$, so the coordinate of $L$ is in $N^\perp$, and it is an isotropic vector there.

To emphasize the dependence of $N$ on $F^2$, define:

\begin{align}
\label{eqn:P_definition}
N\left(F^2\right):= \left(\Lambda^2 F^2 \oplus \ov{\Lambda^2 F^2}\right)\cap W_\bR
\end{align}

and let $\Ein^{1}\left(N(F^2)^\perp\right)$ denote the projectivized quadric of null vectors in $N^\perp(F^2)$.
Then we have the identification

\[
\beta(F^2) \isom \Ein^{1}\left(N(F^2)^\perp\right) \subset
\Ein^{1,2}\subset \bP(W_\bR).
\]

We have denoted by $\Ein$ the quadric of null vectors, and more generally we will use the notation $\Ein^{p,q}\subset \bP(\bR^{p+1,q+1})$ for the quadric of null vectors in a space of indefinite signature.

§.§.§ Oriented electrons

Given $\alpha \in F^2\subset V_\bC$ with $\ip{\alpha,\ov{\alpha}}_{I}=0$, the associated real Lagrangian $span(\alpha,\ov{\alpha})$ has a natural orientation given by $\sqrt{-1}\alpha\wedge\ov{\alpha}$.
Changing $\alpha$ by a complex number does not affect the orientation.
Let us therefore denote by $\beta^{+}(F^2)\subset \LGr^{+}(V_{\bR})$ the family of oriented Lagrangians, and call this the oriented electron.

It follows that given an electron in $\LGr(V_\bR)$, its preimage in $\LGr^{+}(V_\bR)$ has two connected components.
This can also be seen in the model $\bP^+(W_\bR)$, where the preimage of the electron is a null quadric in a subspace of signature $(2,1)$.
It is connected in $\bP(W_\bR)$ but has two components in $\bP^+(W_\bR)$.
Since in $W_\bC$, the data of the electron is determined by the Hodge bundle $\cW^{4,0}=\Lambda^2 F^2\cV$, we will write $\beta^+(\cW^{4,0})$ to denote the two connected components of the oriented electron in $\bP^+(W_\bR)$.

§.§.§ Pseudospheres

Starting from a Hodge decomposition of $W_\bC$, let us denote by

\[
\bS^{1,1}(\cW^{0,4}):=\{ w\in W_\bR\colon \ip{w,w}_{I}=1 \text{ and }\ip{w,\cW^{0,4}}_{I}=0 \}
\]

or in other words, the normalized real positive definite vectors that belong to $\cW^{3,1}\oplus \cW^{2,2}\oplus \cW^{1,3}$.
The corresponding projective two-to-one quotient will be denoted by $\bP^{1,1}(\cW^{0,4})$.

Note that $\bS^{1,1}(\cW^{0,4})$ is homeomorphic to $\bS^1\times \bR^1$, carries a pseudo-Riemannian metric of signature $(1,1)$, and its two “ends” on the null quadric are precisely the two components of the oriented electron.
Furthermore, we have the containment

\[
\bS^{1,1}(\cW^{0,4})\subset \bS^{1,3}(W_{\bR})=\{w\in W_{\bR}\colon \ip{w,w}_I=1\}
\]

and we will denote by $\bP^{1,3}(W_{\bR})$ the two-to-one projective quotient.

§.§.§ Symplectic–Orthogonal dictionary

The next table summarizes the constructions we introduced above.

[c]C0.45 C0.54

Symplectic                Minkowski

$V_\bR \isom \bR^4$     $W_\bR = \Lambda^2_\circ (V_\bR) \isom \bR^{2,3}$

$\Sp_4(\bR)$     $\SO_{2,3}(\bR)$

$\bbU(2)$           $S(\bbO_2(\bR)\times \bbO_3(\bR))$

$\bbU(1,1)$         $S(\bbO_{2,1}(\bR)\times \bbO_2(\bR))$

Lagrangian Grassmannian              Einstein universe

$\LGr(V)$                            $E^{1,2}\subset \bP(W)$

Quadric of isotropic vectors

Line $l\subset V \Rightarrow$ Flag $l\subset l^\perp \subset V$     Isotropic $2$-plane $l\wedge l^\perp \subset W$

$\phi(l)\subset \LGr(V)$                                            $E^{1,2}\supset \bP(l\wedge l^\perp)\isom \phi(l)$

$V_{\bC} = F^2\oplus \ov{F^{2}}$                                    $W_{\bC} = \underbrace{\left(\Lambda^2 F^2 \oplus \ov{\Lambda^2F^2}\right)}_{N_{\bC}} \oplus N^{\perp}_{\bC}$

signature of $F^2$ is $(1,1)$                                       signature of $N$ is $(0,2)$, of $N^\perp$ is $(2,1)$

$\beta(F^2)\subset \LGr(V_{\bR})$                                   $E^{1,2} \supset \Ein^1(N^{\perp})\isom \beta(F^2)$

$1+2+1$ dominated                    $2+1+2$ dominated

The last row, related to domination, will be clarified after <ref>.

§.§ Developing maps to real homogeneous spaces

In this section we show that an assumption A VHS can be used to uniformize domains of discontinuity associated to its monodromy.
One domain is an open set in the real Lagrangian Grassmannian, while another is the full pseudosphere of unit vectors in $W_{\bR}$.

§.§.§ Setup

We keep the notation from <ref> and <ref>, and will make use of the concepts introduced in <ref>.
In particular, $\wtilde{X}$ is the universal cover, we have a variation of Hodge structure $\bW\to X$ of weight $4$ and Hodge numbers $(1,1,1,1,1)$, and it satisfies assumption A from <ref>.
In the discussion below we will pretend that $\bW$ is the (reduced) second exterior power of a weight $3$ variation $\bV$, but this is not necessary.
In order for it to be the case, one might have to pass to a finite cover of $X$, so that the monodromy representation can be lifted from $\SO_{2,3}^{\circ}(\bR)$ to its spin double cover $\Sp_{4}(\bR)$.
All the constructions below can be done while speaking of the variation $\bW$, but we will keep $\bV$ to clarify some of the geometric concepts.
To fix further notation, let

\[
\rho_V\colon \pi_1(X)\to \Sp(V_{\bR})
\quad
\rho_W\colon \pi_1(X)\to \SO(W_{\bR})
\]

be the monodromy representations of the variations of Hodge structure.

In the context of the formula for Lyapunov exponents, in [26] a real Lagrangian is called “bad” if its complexification $L_\bC$ intersects the middle piece of the Hodge filtration $F^2(x)$ at some point $x\in \wtilde{X}$.
For a fixed $F^2(x)$, the collection of all “bad” real Lagrangians is the electron $\beta(F^2(x))$.

Define $\wtilde{Bad}\to \wtilde{X}$ to be the bundle whose fiber above $x\in \wtilde{X}$ is $\beta(F^2(x))$.
The bad bundle is naturally a subset $\wtilde{Bad}\subset \wtilde{X}\times \LGr(V_{\bR})$, which is $\pi_1(X)$-invariant when the action on the second factor is via the representation $\rho_V$.
This invariance implies that $\wtilde{Bad}$ descends to a bundle

\[
Bad:=\left(\leftquot{\pi_1(X)}{\wtilde{Bad}}\right) \to \left(\leftquot{\pi_1(X)}{\wtilde{X}}\right) = X
\]

Let $\wtilde{PS}\to \wtilde{X}$ be the bundle whose fiber above $x\in \wtilde{X}$ is $\bP^{1,1}(\cW^{0,4}(x))\subset \bP(W_\bR)$, see <ref>.
There is a natural inclusion $\wtilde{PS}\subset \wtilde{X}\times \bP(W_{\bR})$ and since $\wtilde{PS}$ is a $\rho_W$-equivariant bundle, it descends to a bundle $PS\to X$.

§.§.§ Developing maps

Define the Lagrangian developing map by projecting the bad bundle to the Lagrangian Grassmannian factor

\[
\wtilde{Bad} \xrightarrow{\Dev^{L}} \LGr(V_{\bR}).
\]

The map $\Dev^{L}$ is by construction $\pi_1(X)$-equivariant.

Similarly, define the “positive” developing map by projecting the pseudosphere bundle to the projective factor:

\[
\wtilde{PS}\xrightarrow{\Dev^P} \bP(W_\bR)
\]

Recall also that $\bP^{1,3}(W_{\bR})$ denotes the projectivized positive-definite vectors, and so $\Dev^P$ lands there.

§.§.§ Oriented versions

We can also decorate the above constructions by taking the double covers that correspond to a choice of relevant orientation.
The corresponding objects will be denoted by a $+$ superscript, or by replacing $\bP$ with $\bS$ where appropriate.

Remark that if $\bW$ is the (reduced) second exterior power of $\bV$, then we have a preferred choice of connected component in $Bad^+$, as explained in <ref>.

* There exists an open set $\Omega_{L}\subset \LGr(V_\bR)=\Ein^{1,2}(W_\bR)$ such that $\Dev^L$ is a bijective diffeomorphism between $\wtilde{Bad}$ and $\Omega_L$.

* Similarly, if $\bW=\Lambda^2_{\circ}\bV$, then the preimage $\Omega^+_L$ of $\Omega_L$ in $\LGr^+(V_\bR)$ has two components, and $\Dev^{L,+}$ is a bijection onto one of them.

* The action of the monodromy group on $\Omega_L$ is properly discontinuous and the quotient is diffeomorphic to the total space of the bundle $Bad\to X$.
This, in turn, is diffeomorphic to the unit circle bundle of the complex line bundle $\cW^{3,1}\to X$, or a two-to-one quotient thereof, depending on whether the real part of the Hodge bundle $\cW^{2,2}$ is topologically trivial, or not.

* The developing map $\Dev^{P}$ is a bijective diffeomorphism between $\wtilde{PS}$ and $\bP^{1,3}(W_{\bR})$.

* The same holds for $\Dev^{P,+}$, $\wtilde{PS}^+$, and $\bS^{1,3}(W_{\bR})$.

* The action of the monodromy group on $\bS^{1,3}(W_{\bR})$ is properly discontinuous, and the quotient is diffeomorphic to the total space of the bundle $PS^+\to X$.
This, in turn, is diffeomorphic to the $\bC^{\times}$-bundle of nonzero vectors of the complex line bundle $\cW^{1,3}\to X$.

As will be proved later on, in <ref>, the open set $\Omega_L$ appearing in <ref> is a domain of discontinuity coming from the log-Anosov property of the monodromy representation.
These further properties will allow us to also identify the complement $\Lambda_L=\LGr(V_{\bR})\setminus \Omega_L$ with a limit set of the Anosov representation.
As will be clear from the proof below, the information in <ref> already implies that for instance the “MUM Lagrangian” in the case of hypergeometric equations, see <ref>, is in $\Lambda_L$.

Both $\LGr(V_\bR)$ and $\wtilde{Bad}$ are real $3$-dimensional manifolds.
The developing map is clearly smooth.
We will establish below that it is also a local diffeomorphism (<ref>), i.e. has invertible differential.
Let us check that it is also globally injective.
Indeed, with notation from <ref>, a null vector $w\in W_{\bR}$ determines a function $f_{w}$ on $\wtilde{X}$, as well as a Lagrangian $[w]\in \LGr(V_\bR)=\Ein^{1,2}(W_{\bR})$.
Scaling $w$ only multiplies $f_w$ by a positive scalar.

Now from the definition of $f_w$, it vanishes at $x\in \wtilde{X}$ if and only if $w^{0,4}(x)=0$, which happens if and only if $[w]\in \beta(\cW^{0,4})$, i.e. $[w]$ belongs to the corresponding electron.
By <ref>, since $w$ is a null vector, $f_w$ can have at most one zero.
Therefore, it is in the image of $\Dev^L$ for at most one point.

The identification of $\wtilde{Bad}$ and the image of $\Dev^L$, denoted $\Omega_L$, is now clear.
Proper discontinuity of the action on $\Omega_L$ follows since the identification is equivariant, and the action on $\wtilde{Bad}$ is visibly properly discontinuous since that is the case for the action of $\pi_1(X)$ on $\wtilde{X}$.

It remains to identify the bundle $Bad$ with the appropriate circle bundle.
It suffices to do so on the lift to the universal cover $\wtilde{Bad}$, in an equivariant way.
First, by possibly passing to a double cover of $X$, we can assume that $\cW^{2,2}$ is a trivial bundle and has a preferred trivializing section.
A double cover suffices since $\cW^{2,2}$ has a real structure and compatible inner product which yields a zero curvature metric.

Now, given a Hodge decomposition

\[
\cW^{4,0}(x)\oplus \cW^{3,1}(x) \oplus \cW^{2,2}(x)
\oplus \cW^{1,3}(x) \oplus \cW^{0,4}(x)
\]

we can parametrize the real null vectors orthogonal to $\cW^{0,4}$ by the unit circle bundle of $\cW^{1,3}$ by:

\[
w = \tfrac{1}{\sqrt{2}}\ov{z} \oplus 1 \oplus \tfrac{1}{\sqrt 2}z \quad \text{ for } z\in \cW^{1,3} \text{ and }\norm{z}^2=1
\]

where $1$ denotes the fixed unit norm trivializing section of $\cW^{2,2}$.

The proof is completely analogous to the previous one.
The only difference is that if $w$ is a positive definite vector in $W_{\bR}$, then by <ref> the function $f_w$ has precisely one zero, so $\Dev^{P}$ is not only injective but also surjective.

To exhibit the identification of the fibers with Hodge data, we proceed analogously, although the formulas are slightly different:

\begin{align}
\label{eqn:developing_map_pseudosphere}
w := \frac{\ov{z}}{\sqrt{2}\norm{z}} \oplus
\frac{\norm{z}^2-1}{\norm{z}^2+1} \oplus
\frac{z}{\sqrt{2}\norm{z}} \quad \text{ for } z\in \cW^{1,3}\setminus \{0\}.
\end{align}

Clearly the $\cW^{3,1}$ and $\cW^{1,3}$ components have norm $\tfrac{1}{2}$ each, and the middle component in $\cW^{2,2}$ has norm strictly bounded by $1$.
The formula is obtained by normalizing any positive-definite vector to have $\cW^{3,1}$ component of norm $\tfrac 12$, and then identifying the interval $(0,\infty)$, to which $\norm{z}$ belongs, with $(-1,1)$ by a succession of $\log$ and hyperbolic tangent.

§.§.§ The developing map is a local diffeomorphism

It remains to verify that $\Dev^L$ and $\Dev^P$ are local diffeomorphisms (as opposed to merely injective maps).
We will only present the calculations in the case of $\Dev^P$, the case of $\Dev^L$ being completely analogous.
When computing the relevant differential, it will also be the case that the matrix of interest will be block-triangular, and in fact the differential of $\Dev^L$ could be identified with a block-diagonal subpart.
Furthermore, the case of $\Dev^L$ is treated by <cit.> in a slightly more general setting, but it also involves a global argument, since their derivative ends up having an extra term coming from a more general Higgs field.
In the case of variations of Hodge structure the corresponding term vanishes, so the result is a purely local calculation.

§.§.§ Explicit trivialization of the developing map

We work locally on $\wtilde{X}$, equipped with a coordinate $z$.
Let also $\alpha(z)$ be a local holomorphic section of $\cW^{1,3}$ and define $s(z):=\frac{\alpha(z)}{\norm{\alpha(z)}}$ to be the associated unit-norm section.
Let also $s_{2,2}$ be a local trivializing, unit norm, section of $\cW^{2,2}$.
Then the developing map can be described locally as

\begin{align}
\label{eqn:local_dev_map_formula}
\begin{split}
\wtilde{X}\times (-1,1) \times \bT^1
& \xrightarrow{\Dev^S}
\bP^{1,3}(W_{\bR})\\
(z,t,\theta) & \mapsto
e^{-\sqrt{-1}\theta} \frac{\ov{s(z)}}{\sqrt{2}} +
t\cdot s_{2,2} +
e^{\sqrt{-1}\theta} \frac{s(z)}{\sqrt{2}}
\end{split}
\end{align}

Note that the three terms on the right are the coordinates with respect to the Hodge decomposition at $z$ in positions $(3,1),(2,2),(1,3)$.
We will omit the superscript $S$ and just write $\Dev$ from now on, and also write $\bP^{1,3}$ instead of $\bP^{1,3}\left(W_{\bR}\right)$.

Let us introduce some further notation.
Work in a neighborhood of a point $z_0\in \wtilde{X}$.
Using the Hodge decomposition at $z_0$, we can write:

\begin{align*}
N = N(z_0) & :=\left(\cW^{4,0}\oplus \cW^{0,4}\right)\cap W_{\bR}\\
P = P(z_0) & :=\left(\cW^{3,1}\oplus \cW^{1,3}\right)\cap W_{\bR}\\
l = l(z_0) & :=\cW^{2,2}\cap W_{\bR}
\end{align*}

so that $W_{\bR} = N \oplus P \oplus l$ which will be our basic decomposition with respect to which we will further decompose vectors and linear maps.

Consider now a point $[w_0]\in \bP^{1,3}$ in the image of the fiber over $z_0$, i.e. $[w_0]=\Dev(z_0,t_0,\theta_0)$ for some $(t_0,\theta_0)$.
By changing $\alpha(z)$ by a complex scalar of norm $1$, and to simplify notation, we can assume that $\theta_0=0$.
Introduce the subspaces

\begin{align*}
P' & =P\cap w_0^{\perp} \text{ which is of the form} \left(\sqrt{-1}\cdot \ov{s(z_0)}+ 0+ \sqrt{-1}\cdot s(z_0)\right)\\
l' & = \text{vectors of the form }\left(\frac{t_0}{\sqrt{2}}\ov{s(z_0)} + s_{2,2} + \frac{t_0}{\sqrt{2}}s(z_0)\right)
\end{align*}

where we have included the components with respect to the three pieces of the Hodge decomposition separately.
Computing the dot products, it follows that

\[
[w_0]^{\perp} \isom N \oplus l' \oplus P'
\]

and so:

\[
T_{[w_0]}\bP^{1,3} = \Hom([w_0],[w_0]^{\perp}) = \Hom([w_0], \oplus l' \oplus P' ).
\]

§.§.§ The derivative of the developing map

We now have a decomposition of the tangent space on the source of the developing map, and on the target, so the differential can be decomposed (writing $\bI:=(-1,1)$):

\[
T_{z_0}\wtilde{X} \times T_{t_0}\bI \times T_{0}\bT^1
\xrightarrow{D_{(z_0,\theta_0,t_0)}\Dev^S}
\Hom([w_0],N\oplus l' \oplus P')
\]

To simplify notation, we will write $D$ for $D_{(z_0,\theta_0,t_0)}\Dev^S$ and omit the $\Hom([w_0],-)$ in the decomposition on the right.

It is now immediate to see from <ref> that the differential is lower-triangular:

\[
\begin{bmatrix}
D\left(T_{z_0}{\wtilde{X}}\to N\right) & 0 & 0\\
D\left(T_{z_0}{\wtilde{X}}\to l'\right) & D\left(T_{t_0}\bI \to l'\right) & 0\\
D\left(T_{z_0}{\wtilde{X}}\to P'\right) & D\left(T_{t_0}\bI \to P'\right) & D\left(T_{0}\bT^1\to P'\right)
\end{bmatrix}
\]

Indeed differentiating <ref> in the $\theta$-parameter only changes the $P$-coordinate of the image, and moves it in the $P'$-direction.
Similarly, differentiating in the $t$-direction gives a non-trivial variation in the $l'$-direction.
So it remains to check that $D(T_{z_0}\wtilde{X}\to N)$ is nondegenerate.
As we will see in a moment, this is precisely given by the second fundamental form $\sigma_{1,3}$, which is the same as $\sigma_{4,0}^{t}$ and is nondegenerate by assumption A.

§.§.§ Differentiating using the flat connection

To compute the needed differential, we apply the Gauss–Manin connection to the individual Hodge components of $\Dev^S$ as defined in <ref>.
We can drop any terms which belong to $P\oplus l$, to find (recall that $s$ is a section of $\cW^{1,3}$):

\begin{align*}
\nabla^{GM}s & = \cancel{\nabla^{Ch}s} + \sigma_{1,3} s + \cancel{\sigma_{2,2}^{\dag}s}\\
& = \sigma_{1,3}s \mod P\oplus l
\end{align*}

Similarly we find

\begin{align*}
\nabla^{GM}\ov{s} & = \sigma_{4,0}^{\dag}s \mod P\oplus l \\
\intertext{and }
\nabla^{GM}s_{2,2} & = 0 \mod P\oplus l
\end{align*}

So we find that the induced map $T_{z_0}\wtilde{X}\to N$ is

\[
v\mapsto \sigma_{1,3}(v)s + \ov{\sigma_{1,3}(v)s} \in N = \left(\cW^{4,0}(z_0) \oplus \cW^{0,4}(z_0)\right)\cap W_{\bR}
\]

In particular it is nondegenerate, and in fact if we equip $T_{z_0}\wtilde{X}$ and $N$ with their natural complex structures (on $N$ from the Hodge decomposition) then the isomorphism is conformal.

It is in fact also possible to avoid altogether local trivializations in the above computations.
Indeed, on the total space of $\cW^{1,3}\to \wtilde{X}$, with zero section removed, there is a natural decomposition of the tangent space coming from the Chern connection.
One can then decompose the differential as in <ref> and write each block of the matrix in terms of naturally defined operators, without the need to choose a trivialization.

§ HYPERGEOMETRIC LOCAL SYSTEMS

For general information on hypergeometric equations, see Beukers–Heckman [9] or Yoshida's book [67].

In this section we show that a large class of examples that satisfy assumption A arises from hypergeometric equations.
The classification of those hypergeometric parameters is accomplished in <ref> and amounts to essentially a local calculation.
Indeed the condition can be seen as a “tiling” property which is local, in the spirit of triangle reflection groups arising from hypergeometric equations that must have angles of the form $\pi/N$ for integers $N$.
The local classification, based on the possible degenerations and the “tiling” condition, is in <ref>.

We take up the “tiling” point of view further and in <ref> we describe the Schwarz reflection structure of general hypergeometric equations.
This embeds the monodromy in a reflection group with index $2$, and we provide the explicit matrices in terms of the hypergeometric parameters.

Next, in <ref> we specialize again to the assumption A case, and describe the tiling of a pseudosphere obtained from using Schwarz reflection.
One can reverse the reasoning and obtain many of the global results on the monodromy by starting from the tiling, but we will not pursue this here.

The period map of the VHS on the real axis takes a particularly simple geometric form and we describe it explicitly in <ref> and subsequent paragraphs.
This is of interest since in general the Griffiths transversality conditions impose restrictions on the period map which are hard to understand globally, but as we show here some control coming from the pseudo-Riemannian point of view can be obtained.

Resurrect remark above

§.§ Assumption A for hypergeometrics

§.§.§ Setup

Hypergeometric equations are traditionally defined on $X:=\bC\setminus \{0,1\}$, although as we shall see in some of the examples of interest it will be useful to consider either $0$ or $\infty$ as orbifold points of some finite order.
Let $\alpha_1,\beta_1,\ldots,\alpha_n,\beta_n$ be a list of $2n$ numbers (while we'll aways take them real later, at this stage they can be complex).
Define the differential operator

\[
D_{\alpha,\beta}:=\prod_{i=1}^n \left(D - \beta_i\right)
- z\prod_{i=1}^n (D+\alpha_i)\quad \text{ where } D = z\partial_z.
\]

With these conventions, the exponents (or: Riemann scheme) of the operator are:

* at $0$: $\beta_1,\ldots, \beta_n$

* at $\infty$: $\alpha_1,\ldots, \alpha_n$

* at $1$: $0, 1, \ldots, n-2, \gamma:=(n-1) - \sum_{i=1}^n (\alpha_i + \beta_i)$

The parameters are slightly different from those of [9], specifically $\beta_i^{BH}=1-\beta_i$, and also slightly different from [29] where $\alpha_i^{Fe}=-\alpha_i$.

§.§.§ Monodromy

The solutions of hypergeometric equations give a local system over the base space, and so a representation $\rho\colon \pi_1(X)\to \GL_n(\bC)$.
When the parameters $\alpha_i$ are invariant modulo $1$ under $x\mapsto -x$, and similarly for $\beta_i$, the representation takes real values.
Furthermore, if irreducible (see below) these local systems are (relatively) rigid, i.e. once the conjugacy classes of the monodromy matrices around cusps are fixed, the representation is unique.

§.§.§ Variation of Hodge structure

If the local system is rigid, it underlies a variation of Hodge structure by the results of Simpson [61].
The Hodge numbers were calculated by Fedorov <cit.>.
The recipe is as follows.
We assume that $\alpha_i,\beta_i$ are real, belong to $[0,1)$ and listed in nondecreasing order (this can always be assumed since shifting the parameters by integers does not affect the monodromy).
Assume, for simplicity, that both lists of numbers are symmetric with respect to $x\mapsto 1-x \mod 1$, so the passage between our list of parameters and Fedorov's is unaffected.
Assume, also, that $\alpha_i\neq \beta_j,\forall i,j$: this ensures the local system is irreducible.

Define now $\rho(k):=\# \{j \colon \alpha_j\leq \beta_k\}-k$
In other words, $\rho(k)$ can be computed by reading the list of all the numbers from left to right until reaching $\beta_k$, starting with the value $0$, and every time an $\alpha$-value is encountered adding $1$, and every time a $\beta$-value is encountered subtracting $1$.
Then the Hodge numbers of the underling VHS are given, up to shift, by $h^{p}=\# \rho^{-1}(p)$.

§.§.§ Local exponents

The above algorithm will yield the parameter space of $\alpha_\bullet,\beta_{\bullet}$ for which the Hodge numbers of the rank $4$ local system are $(1,1,1,1)$.
We also need to make sure that assumption A from <ref> is satisfied.
The criterion is provided by Lemma 6.3 of [26], although see some caveats in <ref> below.
The map $\sigma_{2,1}$ of interest to us is denoted in loc.cit. by $\tau_1$.

Specifically, the criterion implies that away from the singular points $\{0,1,\infty\}$ all the $\sigma_{p,q}$ are isomorphisms.
At the point $1$, the monodromy is a rank $1$ unipotent matrix preserving a line but not a $2$-dimensional subspace, so the map $\sigma_{2,1}$ is an isomorphism by the analysis in <ref>, it corresponds to the second diagram in <ref> (see also <ref> below).
We now list in more detail the situations around singular points where assumption A is satisfied locally, depending on the local exponents and including the orbifold case.

Suppose that $0\leq\mu_1\leq\mu_2\leq \mu_4\leq \mu_4<1$ are also symmetric under $x\mapsto-x \mod 1$ and give the local exponents at the origin of the Picard–Fuchs equation of a variation of Hodge structure on the punctured unit disc.
Then there is a sufficiently small neighborhood of the origin on which assumption A is satisfied, perhaps in the orbifold sense i.e. after passing to a finite cover, if and only if we are in the following cases:

\begin{align*}
(0,0,0,0) \text{ or } (\tfrac 12,\tfrac 12,\tfrac 12,\tfrac 12)
(0,0,\mu,1-\mu) \text{ or } \left(\mu,\tfrac{1}{2},\tfrac{1}{2},1-\mu\right)
& \quad \text{any real }\mu\in\left(0,\tfrac{1}{2}\right)\\
\left(
\tfrac{N-(2k+1)}{2N},
\tfrac{N-1}{2N},
\tfrac{N+1}{2N},
\tfrac{N+(2k+1)}{2N}
\right) & \quad
\text{any integers } k\geq 1, N>2k+1.
\end{align*}

We will refer to <ref> and refer to the possibilities for the unipotent part as case 1, case 2, and case MUM, for the second, third, and fourth diagrams of Hodge numbers.
We will now list the possibilities for exponents, in order of increasing nondegeneracy, and assign them respective cases.
Case 1 and case MUM are good, case 2 isn't.

Suppose first that $\mu_1=0$ then by symmetry also $\mu_2=0$.
If $\mu_3=0$ then necessarily $\mu_4=0$, we are in case MUM.
If $\mu_3>0$, we could have $\mu_3=\mu_4=\tfrac 12$.
This leads to case 2 (there's on a finite cover a $2$-dimensional invariant space).
Otherwise $\mu_4=1-\mu_3\neq \mu_3>0$.
This leads to case 1, and note that $\mu_3\in(0,\tfrac 12)$ can be an arbitrary real.

Now suppose that $\mu_1> 0$, so we have $\mu_4=1-\mu_1$.
If $\mu_2=\mu_1$, so $\mu_3=\mu_4$ then we are again in case 2, unless also $\mu_2=\mu_3$, i.e. all are equal leading to exponents $(\tfrac 12,\tfrac 12,\tfrac 12,\tfrac 12)$, which is case MUM (the elliptic part is $-1$, which can be eliminated by passing to a double cover).
If $\mu_2>\mu_1$, we could have $\mu_2=\mu_3=\tfrac 12$, which leads again to case 1 and exponents $(\mu,\tfrac 12, \tfrac 12,1-\mu)$ for any real $\mu\in(0,\tfrac 12)$.

So we are left with the least degenerate case $0<\mu_1<\mu_2<1-\mu_2<1-\mu_1$.
The monodromy is thus an elliptic matrix, so unless it is of finite order we are not in a good case.
When passing to a finite cover of order $N$, the exponents multiply by $N$.
By <cit.> (it actually applies in this case) for assumption A to hold, we must have $N(1-\mu_2)-N(\mu_2)=1$, or equivalently $\mu_2=\frac{N-1}{2N}$.
Furthermore the local system $\bW$ should be trivial after passage to this finite cover, so its exponents should be integers.
This leads to the requirement $N(\mu_2-\mu_1)=k\in \bN$, or equivalently $\mu_2 = \frac{N-(2k+1)}{2N}$ with $N>2k+1$.
So this leads to the final set of good possibilities:

\[
\left(
\frac{N-(2k+1)}{2N},
\frac{N-1}{2N},
\frac{N+1}{2N},
\frac{N+(2k+1)}{2N}
\right)
\]

with the requirements $k\geq 1, N>2k+1$ (in particular $N\geq 4$).

We are now ready to list the hypergeometric equations which satisfy assumption A.

The variation of Hodge structures underling a hypergeometric equation satisfies assumption A if and only if it appears, up to exchanging the roles of $\alpha$ and $\beta$, in either <ref> or <ref> from the introduction.

To obtain the list, we take all the possible good local choices from <ref>, subject to the condition that $\alpha_i\neq \beta_j \quad \forall i,j$ and that the Hodge numbers, as dictated by Fedorov's recipe in <ref>, are $(1,1,1,1)$.
This occurs if and only if, when placing the numbers on the circle, all the $\alpha$'s and all the $\beta$'s form contiguous clusters (the opposite of interlacing, which leads to finite monodromy).

§.§.§ On local exponents and regularity

Let us clarify the following point.
It is asserted in [26], towards the top of pg. 2328, that if the local exponents at a point are $k,k+1,k+2,\ldots, k+n-1$ then the point is regular.
This requires some qualifications, since for instance the differential operator $(D+1)D + z$ has local exponents $-1,0$ at the origin.
It has a single-valued solution $f_0(z)=1+a_1z+\cdots$ corresponding to the exponent $0$, but a multi-valued solution $f_0(z)\log z + f_{-1}(z)$ where $f_{-1}(z)=\tfrac{1}{z} + b_0 + b_1z + \cdots$.

Such a situation can (and does) occur at the point $z=1$ of the hypergeometric equation, where taking the parameters to be $(\alpha_1,\alpha_2,1-\alpha_2,1-\alpha_1)$ and $(\beta_1,\beta_2,1-\beta_2,1-\beta_1)$ yields the exponents at $1$ to be $-1,0,1,2$.
Nonetheless, the monodromy matrix around $1$ is a rank $1$ unipotent preserving a line.

§.§ Schwarz reflection for hypergeometrics

§.§.§ Setup

We return for this section to the general setting of hypergeometric equations, see the notation in <ref>.
Assume that the parameters $\alpha_i,\beta_j$ are all real.
We now explain how to obtain a Schwarz reflection structure for the associated local system.
This will provide further geometric content to the underlying variation of Hodge structure.

§.§.§ Levelt basis

Let us fix a point in the upper half-plane, say $\sqrt{-1}$, which will be the basepoint for the fundamental group and monodromy representation (see however <ref> below).
A key role throughout the discussion below will be played by the open real segments $(\infty,0),(0,1), (1,\infty)$.
Let the generators of the fundamental group be $g_0$, which crosses first $(0,1)$ and then $(\infty,0)$ and $g_{\infty}$ to cross first $(1,\infty)$ and then $(\infty,0)$.
Let also $g_1$ be the path which crosses first $(1,\infty)$ and then $(0,1)$, so that we have the relation $g_0g_1=g_{\infty}$.

Note that relative to the convention in [9], our $g_{\infty}$ is the same but $g_0,g_1$ are inverses of loc.cit.
Set also $a_j:=\exp(\twopii \alpha_j)$ and $b_j:=\exp(\twopii \beta_j)$, again note that our $a_j$ are the same as in loc.cit, while $b_j$ are the complex conjugates.

\begin{align*}
p_A(t):=\prod_{j=1}^n (t-a_j) & = t^n + A_1 t^{n-1} + \cdots + A_n\\
p_B(t):=\prod_{j=1}^n (t-b_j) & = t^n + B_1 t^{n-1} + \cdots + B_n
\end{align*}

the theorem of Levelt, <cit.> gives that the monodromy of the hypergeometric equation can be conjugated to have the following generators along $g_\infty$ and $g_0$:

\begin{align}
\label{eqn:Levelt_matrices_my_convention}
h_{\infty}=A :=
\begin{bmatrix}
0 & 0 & \ldots & 0 & -A_n\\
1 & 0 & \ldots & 0 & -A_{n-1}\\
0 & 1 & \ldots & 0 & -A_{n-2}\\
&   & \ddots &   & \\
0 & 0 & \ldots & 1 & -A_{1}
\end{bmatrix}
\quad
h_0 = B:= \begin{bmatrix}
0 & 0 & \ldots & 0 & -\ov{B_n}\\
1 & 0 & \ldots & 0 & -\ov{B_{n-1}}\\
0 & 1 & \ldots & 0 & -\ov{B_{n-2}}\\
&   & \ddots &   & \\
0 & 0 & \ldots & 1 & -\ov{B_{1}}
\end{bmatrix}
\end{align}

§.§.§ Schwarz reflection

Since we assumed that our parameters $\alpha_i,\beta_j$ are real, it follows that the differential equation has a complex-conjugation symmetry.
Specifically, starting with a basis of solutions near $\sqrt{-1}$, we can analytically continue along a path that crosses the real axis and reaches $-\sqrt{-1}$, and then apply complex-conjugation, i.e. $\tilde{f}(z):=\ov{f(\ov{z})}$ to obtain another set of functions defined near $\sqrt{-1}$.
Because the differential operator has real coefficients, the functions are again a basis of solutions.

If $V_{\bC}$ denotes the complex vector space of solutions near $\sqrt{-1}$, then the mapped just described is a complex anti-linear self-map of $V_{\bC}$.
Furthermore, it is visibly an involution.
The real subspace of solutions fixed by this operation can be described as follows.
Pick a point $p\in (a,b)$, where $(a,b)$ is the open segment of the real axis crossed by the defining path.
Then around $p$, there is a basis of solutions whose power series have real coefficients.
The analytic continuation of these solutions to $\sqrt{-1}$ yields the fixed points of the complex anti-linear involution described.

§.§.§ Matrices of anti-involutions

Let $\cR_A,\cR_B,\cR_C$ be the transformations described in <ref> above, corresponding to crossing along $(1,\infty)$, $(0,1)$, and $(\infty,0)$ on the real axis.
Then we must have

\[
h_0 = B = \cR_C \circ \cR_B \quad
h_\infty = A = \cR_C \circ \cR_A \quad
h_1 = \cR_B \circ \cR_A \quad
\]

Denote by $\cC\colon \bC^n\to \bC^n$ the operation of coordinate-wise complex conjugation.
Since the transformations $\cR_{\bullet}$ are complex anti-linear, there exist complex linear transformations $R_A,R_B,R_C$ such that $\cR_{X}=\cC\circ R_X$.
After conjugation, these can be arranged to be:

\begin{gather}
\label{eqn:reflection_matrices_Levelt}
\begin{align*}
R_A:= \begin{bmatrix}
0 & \ldots & 0 & 1 & -{A_1}\\
0 & \ldots & 1 & 0 & -{A_{2}}\\
& \reflectbox{$\ddots$} &   &   & \\
% 0 & \ldots & 0 & 0 & -{A_{n-1}}\\
1 & \ldots & 0 & 0 & -{A_{n-1}}\\
0 & \ldots & 0 & 0 & -{A_{n}}
\end{bmatrix}
\quad
R_B:= \begin{bmatrix}
0 & \ldots & 0 & 1 & -\ov{B_1}\\
0 & \ldots & 1 & 0 & -\ov{B_{2}}\\
& \reflectbox{$\ddots$} &   &   & \\
% 0 & \ldots & 0 & 0 & -\ov{B_{n-1}}\\
1 & \ldots & 0 & 0 & -\ov{B_{n-1}}\\
0 & \ldots & 0 & 0 & -\ov{B_{n}}
\end{bmatrix}
\end{align*}\\
% \begin{align*}
\text{ as well as }
R_C:= \begin{bmatrix}
0 & 0 & \ldots & 0 & 1 \\
0 & 0 & \ldots & 1 & 0 \\
& & \reflectbox{$\ddots$} &   &   \\
% 0 & \ldots & 0 & 0 \\
0 & 1 & \ldots & 0 & 0\\
1 & 0 & \ldots & 0 & 0
\end{bmatrix}
% \end{align*}
\end{gather}

Clearly $\cR_{C}^2=1$, so let us verify that the same holds for $\cR_{A}$ (the calculation for $\cR_{B}$ is analogous).
Recall that $\alpha_i$ is real, therefore $a_i=\exp(\twopii \alpha_i)$ satisfies $a_i^{-1}=\ov{a_i}$.
The resulting relation for the coefficients $A_i$ becomes:

\begin{align*}
p_A(t) & = \prod (t-a_i) = t^n \prod \left(1-\frac{a_i}{t}\right)\\
& = t^n \left(\prod a_i\right)\prod\left(a_i^{-1}-\frac{1}{t}\right)\\
& = t^n A_n \prod\left(\ov{a_i}-\frac{1}{t}\right) = t^n A_n \cdot \ov{p_A\left(\frac{1}{t}\right)}
\end{align*}

so we find that

\[
A_l = A_n \cdot \ov{A_{n-l}} \quad \forall l=0,\ldots,n \text{ setting }A_0=1
\]

Computing now $R_A\circ \cC \circ R_A$ yields precisely the identity matrix, showing that $\cR_A^2=\id$.
Prove that the above matrices are actually the correct ones.
Perhaps use Levelt basis?

* It would be more intrinsic to work with a fundamental groupoid in the above situation.
Specifically, $\bP^1(\bC)\setminus \{0,1,\infty\}$ can be broken up into the two half-planes with nonzero imaginary part, each of which is contractible.
These would be the two objects of the groupoid.
The generating arrows (and their inverses) are labeled by which one of the real segments $(\infty,0),(0,1),(1,\infty)$ they cross.

* Some of the above discussion is implicit in [9], see e.g. Prop. 6.1 which shows that the dual local system is given by taking the inverse of the parameters $a_i=\exp(\twopii \alpha_i)$ and $b_j=\exp(\twopii \beta_j)$.

* A bit more generally, for the above discussion it suffices to have that the sets $\{\alpha_i\}$ and $\{\beta_j\}$ are invariant under $z\mapsto \ov{z}$.
In this case the differential operator will still have real coefficients and the needed relation among the $A_i,B_j$ still holds.

§.§.§ Eigenvectors

We make a further simplifying assumption, which holds in our examples of interest: assume that the coefficients $A_i$ and $B_j$ are real, i.e. the set of parameters $\alpha_{\bullet}$ is invariant under $x\mapsto -x \mod 1$ and the same for $\beta_{\bullet}$.
In this case, the monodromy group is real, and we can assume that the complex conjugation $\cC$ is with respect to this real structure.
Note also that each of the complex anti-linear transformations $\cR_{A},\cR_{B},\cR_C$ also induces a real structure – given by its set of fixed vectors – which corresponds to solutions of the differential equation which have real Taylor coefficients on the corresponding segments of the real axis.

We next look at the eigenvectors of the matrices $R_A,R_B$ and $R_C$ from <ref>.
It is clear that $R_C$ has $\left\lceil\tfrac{n}{2}\right\rceil$ eigenvalues $+1$, and $\left\lfloor\tfrac{n}{2}\right\rfloor$ eigenvalues $-1$, with eigenvectors which are symmetric, resp. antisymmetric, when flipping the coordinates across the center.
For $R_A$ and $R_B$, there are again $\left\lceil\tfrac{n-1}{2}\right\rceil$ eigenvalues $+1$, and $\left\lfloor\tfrac{n-1}{2}\right\rfloor$ eigenvalues $-1$, which are in fact common eigenvectors with the same eigenvalues, whose span is the subspace of vectors with last coordinate vanishing.
Finally, $R_A$ has an eigenvector

\[
\begin{bmatrix}
A_{n-1} & A_{n-2} & \cdots &  A_{1} &2
\end{bmatrix}^t
\text{with eigenvalue }-A_n.
\]

Note that by assumption $A_n^2=1$, and that we have the basic relation $A_i = A_n A_{n-i}$.
The eigenvector and eigenvalue for $R_B$ is analogous.

include discussion of the bilinear form
I suspect that the action of the $R_\bullet$ is $(-1)^n$-symmetric, i.e. flips the symplectic form but preserves the orthogonal form.
Maybe easier to see at the complex level?

§.§ Polyhedra and reflections

An analysis of the local structure of the period map in the case which is of interest to us is in the paper of Bryant and Griffiths
Using the Schwarz reflection structure, we are also able to get some of its global properties as well.
Specifically, we get analogues of hyperbolic triangles, in this case polyhedra in the pseudo-sphere $\bS^{1,3}$.
The tiling condition, analogous to that in the polyhedron theorem, is essentially equivalent to assumption A.

§.§.§ Setup

We specialize the discussion in <ref> to the case $n=4$ and with parameters as in <ref>, i.e. satisfying assumption A.
Then all three reflections have the same structure: they have $2$-dimensional eigenspaces for the two eigenvalues $\pm 1$, and the eigenspaces are Lagrangian for the symplectic form.
We therefore focus on one of them, say $R_B$.

Pick a basis $e_1,e_2,f_1,f_2$ of $V_{\bR}$ such that for the symplectic pairing we have $\ip{e_i,f_i}_I=1$ and $0$ otherwise.
Assume also that

\[
R_B e_i = -e_i \text{ and }R_B f_i = f_i
\]

and take a basis of the reduced second exterior power $W:=\Lambda^2_{\circ}V$:

\begin{align}
\label{eqn:exterior_power_basis_coordinates_eigenvalues}
\begin{matrix}
e_1 \wedge e_2 & e_1 \wedge f_2 & e_1\wedge f_1 - e_2 \wedge f_2
& f_1 \wedge e_2 & f_1\wedge f_2\\
a & b & c & d & e \\
+1 & -1 & -1 & -1 & +1
\end{matrix}
\end{align}

where underneath we have written the corresponding names of the coordinate functions for an element of the vector space, and below them the eigenvalues of the matrix $\Lambda^2_{\circ} R_B$.
So the quadratic form on $W$ is (up to a factor of $\tfrac 12$):

\[
Q(w,w):=-a\cdot e + b\cdot d - c^2.
\]

Note that the signature of $Q$ on $W_{\bR}$ is $(2,3)$.

§.§.§ Period map on the real axis

Recall that we are working in a neighborhood $U$ of the segment $(0,1)$ on the real axis.
Recall also that there are two variations of Hodge structure, $\bV$ and $\bW:=\Lambda^2_{\circ}\bV$.
We will describe the middle element of the Hodge filtration, i.e. $F^2\cV$, which is the same as to describe $\cW^{4,0}$.
Viewing it as a line in $W_{\bC}$, this gives a holomorphic map

\[
\gamma\colon U \to \bP(W_{\bC})
\]

which furthermore is equivariant for the action of complex conjugation and $\cR_{\cB}$:

\[
\gamma(\ov{z}) = \cR_{B} \left(\gamma(z)\right)
\]

where recall that $R_{\cB}=\cC\circ R_B$, where $\cC$ is complex conjugation of all coordinates.
Restricting to the real axis, and calling the coordinate $t$, we find

\[
\gamma(t) = \cC \left(R_{B}\cdot \gamma(t)\right)
\]

Let us lift $\gamma$ to a map to $W_{\bC}$, with coordinates $(a:b:c:d:e)$, to find, using the information on the eigenvalues of $R_B$ from <ref>:

\[
a = \ov{a} \quad
b = -\ov{b} \quad
c = -\ov{c} \quad
d = -\ov{d} \quad
e = \ov{d}
\]

or equivalently:

\[
\gamma = \left(a:\sqrt{-1}b:\sqrt{-1}c:\sqrt{-1}d:e\right) \text{ for }a,b,c,d,e \text{ real-valued functions.}
\]

Let us note that it is straightforward to read off the Lagrangian subspace in $V_{\bC}$ parametrized by $\gamma$, or equivalently factorize the vector with coordinates as above:

\begin{align}
\label{eqn:gamma_factorization}
\gamma =\frac{1}{a} \left[
a\cdot  e_1 + \sqrt{-1}\left(d\cdot  f_1 + c\cdot f_2\right)
\right]
\wedge
\left[
a\cdot  e_2 + \sqrt{-1}\left(c\cdot f_1 +  b\cdot f_2\right)
\right]
\end{align}

under the assumptions $ae = c^2 - bd = 1$, see <ref> below.

§.§.§ Riemann bilinear relations

To extract further information, we recall the constraints on $\cW^{4,0}$, which we denote by $\gamma$, and its motion:

Lagrangian The line is isotropic, i.e. $Q(\gamma,\gamma)=0$.

Definiteness The line is negative-definite for the hermitian pairing, i.e. $Q(\gamma,\ov{\gamma})<0$.

Griffiths transversality The tangent $\dot{\gamma}$ is also isotropic, i.e. $Q(\dot{\gamma},\dot{\gamma})=0$.

Let us now express these three conditions in terms of the coordinate functions introduced above (keeping in mind the factors of $\sqrt{-1})$:

\begin{align*}
Q(\gamma,\gamma)=0 & \iff -ae-bd + c^2 =0\\
Q(\gamma,\ov{\gamma})<0 & \iff -ae + bd - c^2 < 0\\
Q(\dot{\gamma},\dot{\gamma})=0 & \iff -\dot{a}\dot{e} - \dot{b}\dot{d} + \dot{c}^2 = 0
\end{align*}

By adding the first two equations, it follows that $a\cdot e>0$.
We also have a scaling symmetry, i.e. we can multiply $\gamma$ by any nonvanishing function.
Let us then rescale it by the unique real-analytic function such that $a\cdot e=1$ holds pointwise, and $a>0$.
Then we the above equations can be rewritten as

\begin{align*}
a\, e & = c^2 - b\, d = 1\\
\dot{a}\, \dot{e} & = (\dot{c})^2 - \dot{b}\, \dot{d} \leq 0
\end{align*}

The assertion that the last line is nonpositive follows from differentiating $a\, e = 1$.
In fact, under assumption A (<ref>), the inequality is strict since its vanishing at a point is equivalent to the vanishing of the second fundamental form $\sigma_{4,0}$ at the same point.
We assume for convenience assumption A from now on.

§.§.§ Further analysis of the image of the period map

In the classical case of triangle reflection groups, the constraints of the Schwartz reflection are enough to pin down geometrically the image of the period map along the real axis – it is a piece of the conformal image of a great circle on the Riemann sphere.
Of course, one has in addition the parametrization of this image, given by hypergeometric functions.

In the present case, the constraints coming from Schwarz reflection do not pin down uniquely the image, but they do give substantial geometric constraints.
Specifically, we can reparametrize the image of $(0,1)$ under $\gamma$ such that $a(t)=\exp(t)$ and $e(t)=\exp(-t)$ for $t\in(t_0,t_1)$ an appropriate time interval (perhaps infinite to one or both sides).
With these normalizations, the only remaining information is in the central three coordinates:

\begin{align}
\label{eqn:period_map_causal_piece}
\begin{split}
c^2 - b\, d & = +1\\
(\dot{c})^2 - \dot{b}\, \dot{d} & = -1
\end{split}
\end{align}

Set $\gamma_1=(b,c,d)$, $\bR^{2,1}$ with coordinates $(b,c,d)$ and quadratic form $Q_1=c^2-bd$, and $\bS^{1,1}:=Q_1^{-1}(1)$.
Then $\gamma_1$ yields a curve on $\bS^{1,1}$ which moves in a timelike manner.
Note that because we normalized $a>0$, the curve is well-defined on $\bS^{1,1}$ and not its two-to-one projective quotient.

§.§.§ Causal structure

A curve on $\bS^{1,1}$ satisfying <ref> is rather heavily constrained.
Let us recall the basic structure of the pseudo-Riemannian spaces $\bS^{p,q}$ which are the unit vectors in $\bR^{p+1,q}$, with the induced pseudo-Riemannian metric $g_{p,q}$.
Let $(\bB^{q},g_q)$ denote hyperbolic space, in its ball model, with associated hyperbolic metric.
Let $(\bS^p,g_p)$ denote the sphere with its standard metric, and $\bS^{p}_{north}$ the open “north” spherical cap, with its induced spherical metric.
Then we have

\[
\left(\bS^{p,q},g_{p,q}\right) \equiv \left(\bS^{p},g_p\right)\rtimes_{\rho}\left(\bB^q,-{g_q}\right)
\approx_{conf} \left(\bS^{p},g_p\right)\times \left(\bS_{north}^q,-g_{q}\right)
\]

Let us explain the notation.
The middle term is a warped product pseudo-Riemannian metric, where the warping function $\rho$ on $\bB^{q}$ is $\left(\frac{1+\norm{\bbx}^2}{1-\norm{\bbx}^2}\right)^2$.
The first identity is an isometry between this warped product and the original $\bS^{p,q}$.
The second identity asserts that this warped product model is conformal, as a pseudo-Riemannian space, to a product between $(\bS^{p},-g_{p})$ and the unit ball viewed as a spherical cap in $\bS^{q}$.
This last identity puts heavy constraints on any $q$-dimensional submanifold of $\bS^{p,q}$ on which the pseudo-Riemannian product is strictly negative-definite: it must be locally the graph of a Lipschitz function $f\colon \bB^{q}\to \bS^p$, and in fact globally so under some completeness assumptions.

The calculations that establish the above identities are contained, for instance, in <cit.> (except that loc.cit. treats $\bH^{p,q}$ and $p=2$, but the calculations are analogous in general).

§.§.§ Behavior at endpoints

For each of the endpoints of the original interval $(0,1)$, there are several possibilities for the behavior, depending on the type of degeneration of the Hodge structure.
We analyze the possibilities, according to the list in <ref>.
Let us also emphasize that there are two kinds of limits we can take:

Line limit The limit of $\cW^{4,0}$ in $\LGr(V_\bC)\subset \bP(W_\bC)$.
This is also affectionately called in the Hodge theory literature the “naive” limit, and will be denoted $\cW^{4,0}_{nv}$.

Plane limit The limit of $\left(\cW^{4,0}\oplus \cW^{0,4}\right)\cap W_{\bR}$ in $\Gr(2;W_{\bR})$, which will be denoted $P^2_{nv}$.

Note that in the corresponding second exterior power of $W_{\bR}$, the coordinates of the vector of interest for the plane limit are

\[
(a:0:0:0:e)\wedge (0:b:c:d:0) \in \Lambda^2 W_{\bR}
\]

since it is determined by the real and imaginary parts of the vector $\gamma$.
In fact, by <cit.>, see also <cit.>, the limits can be computed by looking at the decompositions $I^{p,q}$ of the limit mixed Hodge structure.

Let us note that $\LGr(V_{\bC})$ and $\Gr^+(2;W_{\bR})$ (the double cover of oriented $2$-planes) share some similar features, but also differences, for the action of $G=\Sp_{4}(\bR)$.
The open orbits coincide and give the two copies of Siegel space (depending on orientation/signature) and a copy of $\LGr^{1,1}(V_\bC)$ (Lagrangians where the hermitian pairing has signature $(1,1)$), resp. $\Gr^+_{0,2}\left(W_{\bR}\right)$ ($2$-planes where the quadratic form has signature $(0,2)$).
However, the minimal, closed, orbits are different.
In the Lagrangian Grassmannian it's $\LGr(V_{\bR})$, while in $\Gr^+(2;W_{\bR})$ it is the Grassmannian of isotropic $2$-planes in $W_{\bR}$, which is naturally identified with $\bP^{+}(V_{\bR})$.
As we shall see, the “plane limit”, i.e. in $\Gr^+(2;W_{\bR})$, will be more useful – in <ref> we show that all directions in the boundary of the universal cover have continuous limits in $\Gr^+(2;W_{\bR})$.

Before proceeding to the analysis of degenerations, let us warn that we will refer to the formulas from the proof of <ref>, but the basis $e_i,f_i$ used there, and the one used to express the reflection $R_B$, need not be the same.
So we will extract the conclusions geometrically, in terms of algebraic data associated to the monodromy matrix, instead of referring to a specific basis.
Furthermore, the reflection $R_B$ alone does not determine the unipotent monodromy around the corresponding cusp, so this will lead to some further cases.

§.§.§ Elliptic case

It could be that the Hodge structure does not degenerate at all.
In this case, the line $\cW^{4,0}$ is a fixed point of the corresponding elliptic monodromy matrix and remains in $\LGr^{1,1}(V_{\bC})$.
It also determines a negative-definite two-plane, and these are the line and plane limits.