Non-linear bi-algebraic curves and surfaces in moduli spaces of Abelian differentials

The strata of the moduli spaces of Abelian differentials are non-homogenous spaces carrying natural bi-algebraic structures. Partly inspired by the case of homogenous spaces carrying bi-algebraic structures (such as torii, Abelian varieties and Shimura varieties), Klingler and Lerer recently showed that any bi-algebraic curve in a stratum of the moduli space of Abelian differentials is linear provided that the so-called condition $(\star)$ is fulfilled. In this note, we construct a non-linear bi-algebraic curve, resp. surface, of Abelian differentials of genus $7$, resp. $10$.


Introduction
The study of transcendence properties and unlikely intersections in Diophantine geometry is a fascinating topic possessing a vast literature developing in many directions including those related to the interplay between Hodge theory and the geometry of homogenous spaces such as Abelian varieties and their moduli spaces (that is, Shimura varieties).In this context, an impressive number of heuristic principles and rigorous results 1 was discovered by many authors, and, as a way to unify these statements and also suggest new ones, the point of view of bi-algebraic structures became increasingly popular: see, for instance, the survey [10] and the articles [2] and [3].
In their recent work [9], Klingler and Lerer proposed 2 to extend the bi-algebraic point of view to the non-homogenous setting of moduli spaces of Abelian differentials.More concretely, the moduli space of Abelian differentials of genus g is stratified into complex quasi-projective algebraic orbifolds H(κ) parametrising nontrivial Abelian differentials whose zeroes have multiplicities prescribed by a list κ = (k 1 , . . ., k σ ) such that k 1 + • • • + k σ = 2g − 2. As it was proved by Veech and Masur, the relative periods of the elements of H(κ) can be used to define the so-called period charts inducing a linear integral structure on the analytification of H(κ).In particular, after projectivising the stratum H(κ), κ = (k 1 , . . ., k σ ), we obtain a quasi-projective orbifold H(κ) of dimension 2g − 2 + σ whose analytification possesses a linear projective structure.In this setting, a closed, irreducible, algebraic subvariety W of H(κ) is bi-algebraic if its analytification W an is algebraic in period charts 3 , cf. [9, Def.1.1], and Klingler and Lerer proved that all bi-algebraic curves in the strata H(2) and H(1, 1) of Abelian differentials of genus two are (projectively) linear (in period charts), and, in general, any bi-algebraic curve in H(κ) is linear provided their condition (⋆) is satisfied (cf.[9, Thm.2.8 & Date: June 27, 2023. 1 Such as Ax-Schanuel conjectures, Ax-Lindemann type theorems, André-Oort and Zilber-Pink conjectures. 2 This is a natural goal because the moduli spaces of Abelian differentials seem to "behave" like homogenous spaces: for example, the celebrated breakthroughs by Eskin, Mirzakhani and Mohammadi [5], [6] show that this is the case from the point of view of Dynamical Systems. 3 That is, the relative periods of Abelian differentials projectively lying in W an satisfy exactly codim H(κ) (W ) independent algebraic relations (over C).

2.10]
).Furthermore, they asked whether the bi-algebraicity of a subvariety of H(κ) is already enough to automatically ensure its linearity without extra conditions (cf.[9,Conjecture 2.7]).The main results of this paper say that some bi-algebraic subvarieties can be non-linear: Theorem 1.1.The projectivisation of the family {(C u , ω u )} u∈C\{0,±1} of Abelian differentials defined by is a bi-algebraic curve in H(12) which is not linear.
In particular, observe that if S ⊂ C 3 is a generic algebraic surface, then the projectivisation of the family is a bi-algebraic surface in H(18) which is not linear.We do not have any example of a non linear bi-algebraic subvariety of dimension at least 3 in some stratum of abelian differentials.The proofs of these statements occupy the rest of the paper.More precisely, after a brief discussion of the variations of Hodge structures associated to the families of curves C u and C a,b,c in Section 3, we establish Theorem 1.1, resp.1.3, in Section 5, resp.6.

Eigencohomology of cyclic covers
Let t = (t 1 , . . ., t n ) ∈ C n be a collection of distinct complex numbers.The plane algebraic curve defined by the equation compactifies as a smooth compact Riemann surface C t by adding a = gcd(d, n) points at infinity.The function X t : C t → P 1 defined by X t (x, y) = x is a ramified covering over the sphere having n critical points of order d and a critical point(s) of order d/a.So by Riemann-Hurwitz, the genus g of C t is given by the formula: In particular, if (d, n) = (6, 4), g = 7, and if (d, n) = (6, 5), g = 10.
The covering X t is an abelian covering whose Galois group is generated by the transformation π t (x, y) = (x, ζy) where ζ = exp(2iπ/d).We denote by Unless specified, a holomorphic/meromorphic eigenform on C t is a form η such that π * t η = ζη.The following result is well-known, see, e.g., [12, §3] for more details.We provide the proof for completeness.
Lemma 2.1.In the regime n < d, we have Moreover, the eigenspace where U is a meromorphic function on P 1 , which is holomorphic except possibly at the points t i 's and the point at infinity.In the sequel assume that U does not vanish identically.
Around the point t i , we can write U (x) ∼ cst (x − t i ) ki for some integer k i and a non zero constant.We can also take y as a holomorphic coordinates and we have x − t i ∼ cst y d , so η ∼ (ti,0) cst y dki dy We deduce for η to be holomorphic, k i must be non negative for every i = 1, . . ., n, hence U is a polynomial of x.
It will be convenient to introduce the integers b, c such that n = ab and d = ac.Around a point ∞ of C t at infinity, we have a coordinate z so that x ∼ cst z −c and y ∼ cst z −b .We then have (denoting deg(U ) the degree of U ) ) so we are done.

Finite monodromies
The condition (⋆) of Klingler and Lerer (cf.[9, Def.6.5]) indicates that the families of curves whose Jacobians possess non-trivial fixed parts are potential sources of non-linear bi-algebraic subvarieties: see §6.4 of [9] for more explanations.In this direction, we note that such families of curves were previously studied by several authors in the context of Teichmüller dynamics (e.g., [7], [11], [13], [12], [1] among many others), and, in fact, the families of curves C u and C a,b,c are particular instances of the families of curves described in [12,Thm. 8.3].For later reference, we recall below some of the remarkable properties of the families of Jacobians of C u and C a,b,c .Proof.This statement is included in Theorem 8.3 of [12].Let us provide the proof for completeness.Lemma 2.1 ensures that the Gauss-Manin invariant hermitian form i ) is positive definite, so its monodromy is unitary.Moreover, the bundle 5 are the sole primitive sixth roots of unity).So the monodromy is integral (it is the restriction of the symplectic integral representation to a rational subspace) and preserves a unitary form; hence it is finite.In particular, as it is explained in [12, pp. 929] for instance, since the periods of the corresponding families of forms are holomorphic (multivalued) functions of t which are algebraic when the monodromy is finite, one has that the corresponding families of forms are bi-algebraic.

Picard-Fuchs theory: computation of the Gauss-Manin connection
The vector bundle H 1 ζ over the configuration space B of n distinct complex numbers is isomorphic, thanks to Lemma 2.1, to the product B × C n−2 [x], where C n−2 [x] stands for the space of polynomials of degree ≤ n − 2.More precisely, for the corresponding holomorphic eigenform on C t .The map (t, U ) → ω t,U gives the desired trivialization.In the following, we compute the Gauss-Manin connection on H 1 ζ in this latter: it is defined as follows We denote (4.4) given by Lemma 2.1, the Gauss-Manin connection is given by Proof.Let N i be simply connected neighborhoods of the t i 's, chosen sufficiently small to be disjoint.Denoting by N = ∪N i , and Y = C \ N , the Riemann surfaces and Φ t,t = id.Using this family it is possible to derivate a holomorphic family of holomorphic 1-forms ω t on X −1 t (Y ) with respect to the t kvariables, by the formula ; by shrinking N , this defines a meromorphic 1-form on C t which is holomorphic apart from the points (t i , 0) ∈ C t but that might develop a pole at each (t i , 0) ∈ C t .These poles have no residue at the points (t i , 0), so the period of ∂ t k ω t is well-defined as an element of H This equation is characterized by the existence of a (unique) meromorphic eigenfunction f : C t → P 1 which is such that Here eigenfunction means f • π t = ζf .Let us now make the computations and find the function f .Derivating the multi-valued function y = j (x − t j ) 1/d with respect to the variables t k yields ∂ t k y = −y d(x−t k ) from which we get This meromorphic eigenform has a pole at the point (t k , 0), which in the coordinates y of C t at (t k , 0) has the following local behaviour (assuming that The meromorphic function f : has a unique pole at the point (t k , 0) and it has the following local behaviour (it has no pole at infinity since d ≥ n).In particular, the eigenform with ∇ t k U = V t by (4.5), so it remains to compute (4.10).We have So using the identity Rearranging terms and using P (t k ) = 0 we find the expression of the Gauss-Manin connection given by Proposition 4.1.

Proof of Theorem 1.1
In view of Proposition 3.1, our task is reduced to check that the lines spanned by ω u , u = 0, 1, sit "generically" on H 1 ζ (C u , C).A direct application of Proposition 4.1 shows that the Gauss-Manin on H 1 ζ in the trivialization given by the sections  C) to perform the computations, together with the explicit expression of the Gauss-Manin connection (5.11).We have in this basis Those three vectors are generically linearly independent, and by Leibniz rule, so are the vectors p 2,u , ∇ u p 2,u , ∇ 2 u p 2,u .Remark 5.1.The family ({y 10 = x(x − 1)(x − v)}, [dx/y 9 ]) v =0,1 is a bi-algebraic curve in H(16) because dx/y 9 lies in a fixed part of dimension four (cf.[12,Thm. 8.3]).Nonetheless, a direct calculation using Proposition 4.1 reveals that this bialgebraic curve is linear.

Proof of Theorem 1.3
In view of Proposition 3.1, our task is reduced to check that the lines spanned by ω a,b,c , a, b, c = 0, 1, sit "generically" on the fibers of the vector bundle H 1 ζ .As it turns out, this fact is a particular case of Theorem 6.1 in [12], where a result of Veech and Masur (about the period charts) is cleverly explored to derive that the (period) map taking the configuration of points {0, 1, a, b, c} to [dx/y5 ] is a holomorphic local diffeomorphism. 4n the sequel, we give an alternative proof of the non-linearity of the bi-algebraic surface {(C a,b,c , ω a,b,c )} a,b,c∈C\{0,1} distinct via Proposition 4.1: in particular, we do not rely on the results of Veech and Masur, but rather perform direct calculations (which might be of independent interest).
Proposition 4.1 shows that