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\TopicFR{Géométrie algébrique}
\TopicEN{Algebraic geometry}
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\title[Non-linear bi-algebraic curves and surfaces in \(\mathcal H(12)\) and \(\mathcal H(18)\)] {Non-linear bi-algebraic curves and surfaces in moduli spaces of Abelian differentials}
\alttitle{Courbes et surfaces bi-algébriques non-linéaires dans les espaces de modules de différentielles abéliennes}
\author{\firstname{Bertrand} \lastname{Deroin}}
\address{CNRS \& Universit\'e de Cergy-Pontoise (UMR CNRS 8088), 95302, Cergy-Pontoise, France}
\email{bertrand.deroin@cyu.fr}
\author{\firstname{Carlos} \lastname{Matheus}\IsCorresp}
\address{CNRS \& \'Ecole Polytechnique (UMR CNRS 7640), 91128, Palaiseau, France}
\email{carlos.matheus@math.cnrs.fr}
\begin{abstract}
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.~$10$.
\end{abstract}
\begin{altabstract}
Les strates des espaces de modules de différentielles abéliennes sont des espaces non-homogènes possédant des structures bi-algébriques naturelles. Partiellement inspirés par le cas des espaces homogènes bi-algébriques (comme les tores, les variétés abéliennes et les variétés de Shimura), Klingler et Lerer ont récemment montré qu'une courbe bi-algébrique dans un strate d'un espace de modules de différentielles abéliennes est linéaire pourvu que la soi-disant condition $(\star)$ est satisfaite.
Dans cette note, on construit une courbe, resp. surface, bi-algébrique non-linéaire de différentielles abéliennes de genre $7$, resp. $10$.
\end{altabstract}
\begin{document}
\maketitle
\section{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\footnote{Such as Ax--Schanuel conjectures, Ax--Lindemann type theorems, Andr\'e--Oort and Zilber--Pink conjectures.} was discovered by many authors, and, as a way to unify these statements and also suggest new ones, the point of view of \emph{bi-algebraic structures} became increasingly popular: see, for instance, the survey~\cite{KUY} and the articles~\cite{BKT} and~\cite{BKU}.
In their recent work~\cite{KL}, Klingler and Lerer proposed\footnote{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~\cite{EM}, \cite{EMM} show that this is the case from the point of view of Dynamical Systems.} 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 $H(\kappa)$ parametrising non-trivial Abelian differentials whose zeroes have multiplicities prescribed by a list $\kappa=(k_1,\dots,k_{\sigma})$ such that $k_1+\dots+k_{\sigma}=2g-2$. As it was proved by Veech and Masur, the relative periods of the elements of $H(\kappa)$ can be used to define the so-called \emph{period charts} inducing a linear integral structure on the analytification of $H(\kappa)$. In particular, after projectivising the stratum $H(\kappa)$, $\kappa=(k_1,\dots,k_{\sigma})$, we obtain a quasi-projective orbifold $\mathcal{H}(\kappa)$ of dimension~$2g-2+\sigma$ whose analytification possesses a linear projective structure. In this setting, a closed, irreducible, algebraic subvariety $W$ of $\mathcal{H}(\kappa)$ is \emph{bi-algebraic} if its analytification $W^{\mathrm{an}}$ is algebraic in period charts\footnote{That is, the relative periods of Abelian differentials projectively lying in $W^{\an}$ satisfy exactly $\codim_{\mathcal{H}(\kappa)}(W)$ independent algebraic relations (over $\mathbb{C}$).}, cf.~\cite[Def.~1.1]{KL}, and Klingler and Lerer proved that all bi-algebraic curves in the strata $\mathcal{H}(2)$ and $\mathcal{H}(1,1)$ of Abelian differentials of genus two are (projectively) linear (in period charts), and, in general, any bi-algebraic curve in $\mathcal{H}(\kappa)$ is linear \emph{provided} their condition $(\star)$ is satisfied (cf.~\cite[Thm.~2.8 \& 2.10]{KL}). Furthermore, they asked whether the bi-algebraicity of a subvariety of $\mathcal{H}(\kappa)$ is already enough to automatically ensure its linearity without extra conditions (cf.~\cite[Conj.~2.7]{KL}). The main results of this paper say that some bi-algebraic subvarieties can be non-linear:
\begin{theo}\label{t: non linear bi algebraic curve}
The projectivisation of the family \(\{ (C_u, \omega_{u}) \} _{u\in \C \setminus\{0,\pm 1\}} \) of Abelian differentials defined~by
\begin{equation}\label{eq: curve family}
C_u := \overline{\{ y^6 = x (x-1) (x+1) (x-u)\}} \text{ and } \omega_{u} := x^2\dx/y^5,
\end{equation}
is a bi-algebraic curve in \(\mathcal H(12) \) which is not linear.
\end{theo}
\begin{rema}
Note that \(C_u\) is a branched cover of \(\overline{\{z^2=x(x-1)(x+1)(x-u)\}}\). In particular, the family of Abelian differentials $\{(C_u, [\dx/y^3])\}_{u\in\mathbb{C}\setminus\{0,\pm1\}}$ corresponds to an arithmetic Teichm\"uller curve (in the sense of~\cite[\S 5]{GJ}).
\end{rema}
\begin{theo}\label{t: non linear bi algebraic surface}
Given a generic algebraic curve \(C\subset \mathbb C^3\), the projectivisation of the family \(\{ (C_{a,b,c}, \omega_{a,b,c}) \} _{\substack{(a,b,c)\in C, \\ a,b,c\in \C \setminus\{0, 1\} \\ \text{distinct}}}\) of Abelian differentials defined by
\begin{equation}\label{eq: surface family}
C_{a,b,c} := \overline{\{ y^6 = x (x-1) (x-a) (x-b) (x-c)\}}\quad \text{and}\quad \omega_{a,b,c} := \dx/y^5,
\end{equation}
is a bi-algebraic curve in \(\mathcal H(18) \) which is not linear.
\end{theo}
In particular, observe that if \(S\subset \mathbb C^3\) is a generic algebraic surface, then the projectivisation of the family \(\{ (C_{a,b,c}, \omega_{a,b,c}) \} _{\substack{(a,b,c)\in S, \\ a,b,c\in \C \setminus\{0, 1\} \\ \text{distinct}}}\) is a bi-algebraic surface in \(\mathcal H (18)\) 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~\ref{s.fixed-parts}, we establish Theorem~\ref{t: non linear bi algebraic curve}, resp. \ref{t: non linear bi algebraic surface}, in Section~\ref{s.Thm1}, resp. \ref{s.Thm2}.
\section{Eigencohomology of cyclic covers}
Let \(t=(t_1,\ldots, t_n) \in \mathbb C^n \) be a collection of distinct complex numbers. The plane algebraic curve defined by the equation
\[
y^d = (x-t_1) \ldots (x-t_n)
\]
compactifies as a smooth compact Riemann surface \(C_t\) by adding \(a= \gcd(d,n)\) points at infinity. The function \(X_t: C_t \rightarrow \mathbb P^1 \) 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:
\[
g = \frac{(n-1) (d-1) - (a-1) }{2}.
\]
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 \(\pi_ t (x,y) = (x, \zeta y) \) where \(\zeta = \exp (2i\pi /d)\). We denote by \(H^* _\zeta (C_t, \mathbb C) \) the eigenspace \(\Ker (\pi_t^* - \zeta) \subset H^* (C_t, \mathbb C)\). Unless specified, a holomorphic/meromorphic eigenform on \(C_t\) is a form \(\eta\) such that \(\pi_t^* \eta = \zeta \eta\). The following result is well-known, see, e.g., \cite[\S3]{McM} for more details. We provide the proof for completeness.
\begin{lemm}\label{l: trivialization}
In the regime \(n < d\), we have
\[
H^{1,0} _{\zeta} (C_t,\mathbb C) = \left\{ U(x) \frac{\dx}{y^{d-1}} \,\middle |\, U \text{ polynomial of degree } \leq n-2 \right\}.
\]
Moreover, the eigenspace \(H^1_{\zeta }(C_t, \mathbb C)\) is made of cohomology classes of holomorphic eigenforms, i.e.
\[
H^1_{\zeta} (C_t,\mathbb C)\simeq H^{1,0} _{\zeta}(C_t,\mathbb C) \text{ and } H^{0,1} _\zeta (C_t, \mathbb C) = \overline{H^{1,0}_{\zeta^{d-1}} (C_t, \mathbb C)} =0.
\]
\end{lemm}
\begin{proof}
A form in \(H^{1,0} _{\zeta} (C_t)\) can be expressed as \(\eta = U(x) \frac{\dx} {y^{d-1} } \) where \(U\) is a meromorphic function on \(\mathbb P^1\), 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 \(U(x) \sim \cst (x-t_i) ^{k_i}\) for some integer \(k_i\) and a non zero constant. We can also take \(y\) as a holomorphic coordinates and we have \(x-t_i \sim \cst y^d\), so
\[
\eta \sim _{(t_i, 0)} \cst y^{dk_i } dy
\]
We deduce for \(\eta\) to be holomorphic, \(k_i\) must be non negative for every \(i=1,\ldots, n\), 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 \(\infty\) of \(C_t\) at infinity, we have a coordinate \(z\) so that \(x\sim \cst z^{-c}\) and \(y\sim \cst z^{-b} \). We then have (denoting \(\deg (U)\) the degree of \(U\))
\[
\eta \sim _{\infty} \cst z^{- c \deg (U) - c - 1 +b(d-1) }
\]
which shows that \(\eta\) is holomorphic iff \(\deg (U) \leq \frac{b(d-1) - c - 1}{c}= n - 1- \frac{ (b+1) }{c} \), or equivalently, since \(b