Loci in strata of meromorphic differentials with fully degenerate Lyapunov spectrum

We construct explicit closed GL(2, R)-invariant loci in strata of meromorphic differentials of arbitrary large dimension with fully degenerate Lyapunov spectrum. This answers a question of Forni-Matheus-Zorich.


Introduction
Lyapunov exponents of the Teichmüller flow have been studied a lot since the work of Zorich ([Zor97], [Zor99]) and Forni [For02]. Their understanding is important for applications to the dynamics of interval exchange transformations and polygonal billiards. A big breakthrough is the Eskin-Kontsevich-Zorich formula for the sum of positive Lyapunov exponents [EKZ11b]. Given a SL(2; R) invariant suborbifold of a stratum of quadratic differentials, they relate the sum λ 1 + · · · + λ g to the Siegel-Veech constant of the invariant locus 1 .
By a theorem of Kontsevich and Forni, the sum λ 1 + · · · + λ g is also the integral over the invariant locus of the curvature of the Hodge bundle along Teichmüller discs ( [For02], [EKZ11b]). Using this interpretation, every Lyapunov exponent is computed for cyclic covers of the sphere branched over 4 points ([EKZ11a], 2010 Mathematics Subject Classification. Primary: 30F60, 32G15, 32G20; Secondary: 37H15. 1 For quadratic differentials, two bundles can be considered. In this article, we will only be interested in the bundle with fiber H 1 (X, R) over a Riemann surface X. The Lyapunov exponents of this bundle are often denoted by λ + 1 , . . . , λ + g . [FMZ11], see also [BM10], and [Wri12] for abelian covers). For some cyclic covers, Forni-Matheus-Zorich remarked that the sum λ 1 + · · · + λ g is equal to zero [FMZ11,Thm. 35]. This surprizing fact means that the complex structure of the underlying Riemann surface is constant along the Teichmüller disc. Forni-Matheus-Zorich ask whether it is possible to construct other invariant loci with this property (see [FMZ11,p. 312]). The content of this article is to give a simple explanation of the phenomenon discovered by Forni-Matheus-Zorich. We prove: Theorem 1. There exist closed GL(2; R) invariant loci of quadratic differentials of arbitrarily large dimension with zero Lyapunov exponents.
This result can be interpreted in the following way: the projection of such a locus to the moduli space of compact Riemann surfaces is constant. Remark that the situation for strata of abelian differentials is completely different: there are finitely many invariant suborbifolds with fully degenerate Lyapunov spectrum (meaning in this setting that all exponents are zero except λ 1 which is 1), and they are arithmetic Teichmüller curves (see [Möl11], [For06], [FMZ11] and [Aul13]).

The Teichmüller flow for translation surfaces.
A translation surface is a pair (X, ω) where X is a compact Riemann surface and ω is a holomorphic one-form on X. If S (ω) if the set of the zeroes of ω, there exists an open covering ofX = X \ S (ω) and holomorphic charts ϕ i : U i → X such that ϕ * i ω = dz for all i. For such an atlas, the transition functions are translations. The form ω induces a flat metric |ω 2 | on the open surfaceX, whose area is the integral i 2 X ω∧ω. We could have taken meromorphic 1-forms instead of holomorphic ones, but in that case the area of the surface is never finite.
There is a natural action of GL(2, R) on translation surfaces given as follows: first we pick an atlas of charts ofX where all transitions functions are translations by some complex vectors v i j which we will consider as vectors in R 2 . Then, for any M is GL(2, R), we get an open surfaceX M defined by an atlas whose transition functions are translations by M.v i j . This surface is diffeomorphic tõ X M . Therefore, we can fill the holes and extend the complex structure in a unique way: the result is a compact Riemann surface X M diffeomorphic to X endowed with a meromorphic differential ω M of finite volume, hence holomorphic. The action of GL(2; R) is defined by the formula M.(X, ω) = (X M , ω M ). The action of SL(2; R) preserves the volume of translation surfaces.
If we fix multiplicities (m 1 , . . . , m r ) such that r i=1 m i = 2g − 2, the stratum of translation surfaces H(m 1 , . . . , m r ) is the set of translations surfaces (X, ω) where ω has r distinct zeroes of multiplicities m 1 , . . . , m r modulo diffeomorphism. The normalized stratum H 1 (m 1 , . . . , m r ) is the locus of flat surfaces with unit area in H(m 1 , . . . , m r ), and the projective stratum PH(m 1 , . . . , m r ) is obtained by taking the quotient of H(m 1 , . . . , m r ) under the natural C × -action on forms. Strata and projective strata are complex orbifolds of respective dimensions dimensions 2g + r − 1 and 2g + r − 2 if g ≥ 2.
There are standard coordinates on the stratum H(m 1 , . . . , m r ), called period coordinates. Fix (X, ω) in this stratum, and let A 1 , . . . , A g , B 1 , . . . , B g be a symplectic basis of H 1 (X, Z) and C 1 , . . . , C r−1 be r − 1 paths joining a zero of ω to all the r − 1 other zeroes. The map Assume that the Veech group Γ of (X, ω) is a lattice in SL(2; R). Then the image H/Γ of the corresponding Teichmüller disc in the projective stratum is called a Teichmüller curve.
All these considerations generalize to the so-called half-translation surfaces, which are pairs (X, q) where q is a quadratic holomorphic (for the time being) differential on X. The transitions functions of a well-choosen atlas of charts on the open surface are half translations, that is either translations or flips. The area of the flat metric onX is 1 2 X |q|, and we still have an action of GL(2, R) as well as a Teichmüller flow. The period coordinates on strata of quadratic differentials are obtained as follows: for any (X, q) in a stratum, we take the twofold branched covering p : X → X given by the holonomy representation of q, which is given by a morphism from π 1 (X) to Z/2Z. Let j be the corresponding involution acting on X. The quadratic differential p * q is the square of an abelian differential ω. The period coordinates of (X, q) are obtained by taking J-anti-invariant absolute and relative periods of (X, q).
However, a major difference happens for quadratic differentials: it is possible to take meromorphic quadratic differentials and still get finite volume for the corresponding flat surface. More precisely, (X, q) has finite volume if and only if q has poles of order at most one. Therefore we have strata, normalized strata and projective strata Q(m 1 , . . . , m r ), Q 1 (m 1 , . . . , m r ) and PQ(m 1 , . . . , m r ), where r i=1 m i = 4g − 4 and each m i is either positive or equal to −1. Let S be a finite subset of X with cardinal n, so that (X, S ) gives a point in the marked Teichmüller space T g,n (genus g with n marked points). The cotangent space of T g,n at X is exactly the space Q S (X) of holomorphic quadratic differentials on X\S with poles of order at most one on S. There is a norm on Q S (X) given by ||q|| = X |q|, as well as a dual norm on Q S (X) * . The corresponding distance on T g,n is the Teichmüller metric.
Let us fix (X, S ) as well as an element q in Q S (X). There is a complex linear form µ q on Q S (X) given by scalar product with the L ∞ Beltrami differential |q| q : Note that µ q (q) = X |q| > 0 so that µ q is nonzero. Besides, we have ||µ q || = 1. The map q → µ q gives a non-linear isomorphism between the unit spheres of Q S (X) and Q S (X) * , hence between the unitary cotangent space U * T g,n and the unitary tangent space UT g,n .
If (X, q) is given and S is the set of poles of q, the Teichmüller flow of (X, q) introduced formerly is the geodesic flow (for the Teichmüller metric) on T g,n starting from X in the direction µ q .

The period mapping.
For any compact Riemann surface X, H 1 (X, C) is the orthogonal sum (for the intersection form) of Ω(X) and Ω(X). Besides, the composition is an isomorphism. The Hodge norm || || Hodge is the unique norm on H 1 (X, R) making ψ an isometry.
Let us now consider a local holomorphic family of curves, that is a proper holomorphic submersion π : X → B whose fibers are compact Riemann surfaces of some genus g, where B is a small ball in C n . The Hodge bundle is a holomorphic vector bundle on B of rang g whose fiber at each point b is the vector space Ω(X b ). The local system R 1 π * R X is trivial, which means that we can canonically identify all the vector spaces H 1 (X b , R) to some fixed real vector space V of dimension 2g. The local period map ξ : B → Gr(g, V C ) associates to any b the subspace H b in the Grassmannian of g-dimensional complex subspaces of V C . The derivative of ξ at a point b in B is a linear map from The differential of ξ can be explicitely computed: ξ induces a classifying map ξ Teich : B → T g . Then we have the following formula due to Ahlfors: for any vector v in T b B and any elements α and β in Ω(X b ), In this formula, ξ Teich (v) is a tangent vector to T g , hence represented by a Beltrami differential which is a tensor field on X of type (−1, 1). Thus, the integrand in the above formula of type (2, 0) + (−1, 1) = (1, 1). We can also think of ξ Teich (v) as a linear form on Q(X); in this case the above formula reads It is also possible to give another interpretation on ξ . For this we consider the exact sequence of holomorphic vector bundles The bundle V⊗O B carries a natural flat connection (the Gauß-Manin connection), but H is not in general a flat sub-bundle of V ⊗ O B . A precise way to measure this (see formula (2) below) is the second fundamental form σ associated with this exact sequence and the Gauß-Manin connexion; it is a (1, 0)-form with values in Hom (H, H). A simple calculation shows that The Hodge bundle H carries a natural metric given by the intersection form, its curvature form is given by the formula By "∧" we mean composition on the fiber and wedge-product on the base. In particular, i TrΘ H is a positive (1, 1)-form on B.
For any compact half-translation surface (X, q), Forni's B-form is a bilinear form on Ω(X) defined by If ξ Teich (v) has unit norm, we can write it as µ q for some holomorphic quadratic differential on X. Then we have (β, ξ v (α)) = B q (α, β). In case of a Teichmüller orbit (X t , q t ), if we differentiate along the vector field ∂ ∂ t , we get the formula (3) (β, ξ ∂ t (α)) = B q t (α, β).
Note that the exponents λ i are called λ + i in numerous papers (e.g. in [EKZ11b], [FMZ11]). The exponents λ − i will never be considered in the article. By (4), all λ i 's are at most one. If the component Q is orientable, which means that every quadratic differential occuring in the stratum is the square of an abelian differential, then the top Lyapunov exponent λ 1 equals one. If not, the norm of Forni's B form is strictly smaller than one so that λ 1 < 1.
For any (X, q) in a stratum PQ(m 1 , . . . , m r ), the Poincaré metric on H induces a metric on the Teichmüller disc passing through (X, q). The corresponding volume element defines a relative (1, 1) form dV Teich , where by "relative" we mean relative with respect to the foliation by Teichmüller discs. If Θ is the curvature of the Hodge bundle on PQ(m 1 , . . . , m r ), its trace is also a relative (1, 1) form on the projective stratum. Let Λ : PQ(m 1 , . . . , m r ) → R be defined by the formula

Λ =
Tr Θ dV Teich · Then Kontsevich-Forni's main formula for the Lyapunov exponents is where D is the projection of D 1 in the projective stratum and dV is the normalized volume element on D of total mass one. For any (X, q) in D, let θ 1 , . . . , θ g be the eigenvalues of Forni's B-form in the direction of the Teichmüller flow when diagonalized in an orthonormal basis for the intersection form. Using formulae (2), (1) and (3), we see that Forni's inequality implies that θ i (X, q) ≤ 1 for all i so that λ 1 + . . . + λ g ≤ g.
Thanks to the main result of [EM13], any closed SL(2; R)-invariant locus in the projective stratum PQ(m 1 , . . . , m r ) is affine in period coordinates, hence carries a natural SL(2; R)-invariant probability measure. It is also possible to define Lyapunov exponents for this measure, and formula (5) holds.
If (X, q) is any half-translation surface, the closure of its SL(2; R)-orbit in the normalized stratum is affine in period coordinates. It follows from [CE13] that almost every direction θ, the real Teichmüller flow of (X, e iθ q) is Osseledets-generic for the corresponding natural probability measure. Therefore it makes sense to consider Lyapunov exponents of (X, q), and formula (5) is still valid if we integrate on the closure of the PGL(2; R)-orbit. (i) If X is not hyperelliptic or if g ≤ 2, τ is surjective.
(ii) If X is hyperelliptic, Im (τ) has codimension g − 2 in Q(X) and consists of the quadratic differentials invariant by the hyperelliptic involution.
Since τ is the transpose of the derivative of the period map, Noether's result has the following geometric interpretation: Proposition 2 (Infinitesimal Torelli's theorem, [Voi07,Cor. 10.25]).
Let ξ : T g → H g be the period map. Then ξ is an immersion outside the hyperelliptic locus or everywhere if g ≤ 2, and the restriction of ξ to the hyperelliptic locus is also an immersion.
Remark that Forni's B-form factors through Im τ, and can be extended naturally to Q(X) by the formula B q (q) = Xq |q| q · The key proposition of this section is: Proposition 3. Let (X, q) be a half-translation surface, n the number of poles of q, and D be its Teichmüller disc. Then the following are equivalent: (i) D lies in the determinant locus.
(ii) The forgetful map T g,n → T g maps D to a point.
(iii) For any (X t , q t ) in D, the extension of B q t to Q(X t ) vanishes.
(iv) All Lyapunov exponents of (X, q) are zero. Proof.
(i) ⇒ (ii) Using (3), the composite map D → T g,n → T g τ − → H g has zero derivative. Assume that D is not contained in the hyperelliptic locus. Thanks to the infinitesimal Torelli theorem, D is mapped to a point via the forgetful map T g,n → T g . Assume now that D is contained in the hyperelliptic locus. Then the restriction of τ to this locus is again an immersion, and we can apply the same argument.
(ii) ⇒ (iii) If (X t , q t ) is a point in D, the derivative of projection of the Teichmüller flow of (X t , q t ) on T g is the linear formq → B q t (q) on Q(X t ).
Since all θ i 's are nonnegative and continuous functions, λ 1 = . . . = λ g = 0 if and only if all θ i 's vanish on D.
Corollary 1. If q is a holomorphic quadratic differential on X, the Teichmüller disc of (X, q) is not included in the determinant locus.
Proof. If q is holomorphic, B q (q) > 0 and we apply Proposition 3.
Remark 1. In the hyperelliptic case, it can happen that q is holomorphic but that (X, q) lies in the determinant locus. Let X be an hyperelliptic surface of genus at least 3, let j be the hyperelliptic involution, and let q be an anti-invariant holomorphic quadratic differential (if X is the Riemann surface of a polynomial w 2 − P(z), we can take q = w −1 dz ⊗2 ). Since any holomorphic 1-form on X is anti-invariant under j * , B q = 0. Hence (X, q) lies in the determinant locus, but the Teichmüller disc of (X, q) goes outside of the hyperelliptic locus.
We can give an explicit lower bound on the number n.
Proposition 4. Let (X, q) be a half-translation surface of genus at least 1 satisfying the equivalent conditions of Proposition 3. Then q has at least max (2g − 2, 2) poles.
Proof. The fact that the number n of poles of q must be at least one follows from [Kra81, Thm 4']. To get the lower bound 2g − 2 in the proposition, we use [EKZ11b, Thm 2] for the closure of the SL(2; R)-orbit O of (X, q), which is contained in a stratum PQ (−1) n , m 1 , . . . , m r : we get and we get the required estimate.
Remark 2. We will see that this bound is asymptotically sharp in §3.3.

3.2.
Pillow-tiled surfaces. In this section, we give constraints on pillow-tiled surfaces whose Teichmüller disc lies in the determinant locus. Let us start with a technical result: Proposition 5. Let X be a Riemann surface of genus g, B(t 0 , ε) a small ball in C \ {0, 1, ∞}, and ϕ : X × B(t 0 , ε) → P 1 be a holomorphic map satisfying the following conditions: (1) For any t in B(t 0 , ε), ϕ t is non-constant and B(ϕ t ) = {0, 1, ∞, t}.
(2) The configuration of the ramification points of ϕ t remains constant with t. If d is the degree of the branched coverings ϕ t , then 3(g − 1) ≤ d.
Proof. For any x in X, let s(x) = ∂ ∂t |t=t 0 ϕ t (x) ∈ T ϕ t 0 (x) P 1 . Then s is a holomorphic section of the holomorphic line bundle ϕ * t 0 TP 1 . Let x 0 be a ramification point of ϕ t 0 such that ϕ t 0 (x 0 ) = 0. Let us assume that s(x 0 ) 0. By the implicit function theorem, the equation ϕ t (x) = 0 has a unique solution (x, t(x)) depending holomorphically on x for (x, t) near (x 0 , t 0 ). Since ϕ t(x) (x) = 0, we get By hypothesis, x is a ramification point of ϕ t(x) , i.e. (ϕ t(x) ) (x) = 0. Besides, since ∂ t ϕ(t(x), x) → s(x 0 ) as x → x 0 , t vanishes. Hence ϕ t 0 (x) vanishes for x near x 0 , so that ϕ t 0 is constant and we get a contradiction. It follows that s vanishes at x 0 . The same result also holds over any ramification point of ϕ t 0 lying over 1 and ∞. Lastly, if ψ t (x) = ϕ t (x) − t, the argument we used proves that for any ramification point x of ψ t lying over 0, ∂ ∂t |t=t 0 ψ t (x) = 0, which means that s(x) = 1. In particular s in nonzero.
We can now decompose the ramification divisor R of the branched covering ϕ t 0 as the sum R 0 + R 1 + R ∞ + R t . Besides, we can assume that deg R t is smaller than deg R 0 , deg R 1 and deg R ∞ , otherwise we move the points 0, 1, ∞ and t by a suitable homographic transformation. Besides, thanks to the Riemann-Hurwitz formula, we have deg R = 2(g + d − 1) Now s is a nonzero section of the line bundle L = ϕ * t 0 TP 1 (−R 0 − R 1 − R ∞ ), and The result follows.
Corollary 2. Let (X, q, π) be a pillow-tiled surface of genus g, and let d be the degree of π. If the Teichmüller disc of (X, q) lies in the determinant locus, then d ≥ 3(g − 1).
Remark 3. It is not possible to find an upper bound on the primitive degree d in a given connected component of strata since there are infinitely many pillow-tiled surfaces with arbitrary large primitive degree.
Let (X, q) be a half-translation surface and (Y, π) be an arbitrary finite covering of X with branching locus S. Assume that for any point y in Y above S, the ramification index of π at y is at least 2. Then π * q is holomorphic, so that B π * q is non zero on Q(Y). Thanks to Corollary 1, the Teichmüller disc of (Y, π * q) doesn't belong to the determinant locus. Using this observation, we can prove the following: Corollary 3. Let (X, q) be a half-translation surface and (Y, π) be a finite Galois covering of X with branch locus S. If the Teichmüller disc of (Y, π * q) lies in the determinant locus, then at least one pole of q does not belong to S.
As a particular by-product, we get: Proposition 6. Let (X, q, π) be a pillow-tiled surface such that π is Galois. Then the Teichmüller disc of (X, q) lies in the determinant locus if and only the branching locus of π contains at most three points.
Proof. Let q st be the standard meromorphic differential on P 1 with four simple poles such that q = π * q st . Then the branching locus of π lies in the set of poles of q st . If X is in the determinant locus, according to Corollary 3, one of the poles of q st is not a branching point of π.
Conversely, assume that the branching locus of π has less than four points. If {z 1 , z 2 , z 3 , z 4 } are the four poles of q st , let us assume that z 4 is not a branch point of π. The complex Teichmüller flow of (P 1 , q st ) is of the form (P 1 , q t ) where q t has poles at z 1 , z 2 , z 3 and another point z 4 (t) such that [z 1 , z 2 , z 3 , z 4 (t)] = t. Let • X be the open Riemann surface obtained by removing π −1 {z 1 , z 2 , z 3 }. Then • X is an unramified covering of P 1 \ {z 1 , z 2 , z 3 }. It follows that ( • X \ π −1 (z 4 (t)), π * q t ) parametrizes the Teichmüller disc of (X, q) in T * T g,n (where n is the number of poles of q). This disc maps to {X} via the forgetful map T g,n → T g . Thanks to Proposition 3, the Teichmüller disc of (X, q) lies in the determinant locus.
Let us now consider pillow-tiled surfaces arising as cyclic coverings of the projective line. They are given by a combinatorial datum (N, a 1 , a 2 , a 3 , a 4 ) where 0 < a i ≤ N, gcd (a 1 , a 2 , a 3 , a 4 , N) = 1 and 4 i=1 a i ≡ 0 (N): the associated cyclic covering is the Riemann surface of the polynomial w N − (z − z 1 ) a 1 (z − z 2 ) a 2 (z − z 3 ) a 3 (z − z 4 ) a 4 .
In topological terms, if (γ i ) 1≤i≤4 are small loops around the z i 's for 1 ≤ i ≤ 4, then the kernel of the group morphism π 1 (P 1 \ {z 1 , z 2 , z 3 , z 4 }) → Z/NZ given by γ i → a i defines a true cyclic covering of P 1 \ {z 1 , z 2 , z 3 , z 4 } of degree N, which extends to a branched cyclic covering of the projective line.
In [FMZ11,Thm. 35], the authors prove that all Lyapunov exponents of the Teichmüller curve corresponding to a cyclic covering are 0 if one of the integers a i equals N.
Proposition 7. If (X, q) is a pillow-tiled surface obtained by a cyclic covering of P 1 with combinatorial datum (N, a 1 , a 2 , a 3 , a 4 ), then the Teichmüller disc of (X, q) lies in the determinant locus if and only if one of the a i s equals N.
Proof. Thanks to Proposition 6, it suffices to prove that the projection π of the covering is branched at three points or less if and only if one of the a i s equals N. If {z 1 , z 2 , z 3 , z 4 } are the four points defining the cyclic cover, the ramification index of π at any point of π −1 (z i ) is N pgcd (N,a i ) qed.

Construction of invariant subvarieties.
In this section, we provide the precise statement underlying Theorem 1 as well as its proof.
Let m 1 , . . . , m r and k be positive integers such that ( r i=1 m i ) − k = −4, and let S be the set of couples (q, x 1 , . . . , x k−3 ) such that such that q is a meromorphic differential on P 1 with simple poles at 0, 1 and ∞ and the x i 's, and q has r zeroes of order m 1 , . . . , m r . It is a smooth GL(2; R)-invariant submanifold of T * M 0, [k] (where the bracket means that the points are ordered).