Ax–Schanuel conjecture for strata of abelian differentials

Let $\textnormal{S}_{\alpha}$ be a stratum of abelian differentials up to scaling, let $\widetilde{\textnormal{S}_{\alpha}^{\mathrm{an}}}$ be the universal cover of its analytification, and let $V\subset\widetilde{\textnormal{S}_{\alpha}^{\mathrm{an}}}$ be an irreducible algebraic subvariety, meaning a closed irreducible analytic subvariety algebraic in the period charts. Let $\pi$ denote the projection to $\textnormal{S}_{\alpha}$. Ax–Schanuel conjecture for $\textnormal{S}_{\alpha}$. The Zariski closure $\overline{\pi(V)}^{\mathrm{Zar}}$ of the projection of $V$ in $\textnormal{S}_{\alpha}$ is bi-algebraic in $\textnormal{S}_{\alpha}$, hence linear in view of the bi-algebraic linearity conjecture.

This is identified as the main geometric step toward the proposed Zilber–Pink conjecture for strata, paralleling Ax–Schanuel statements for other bi-algebraic structures. The particular case stated here is presented as a conjecture and remains open in general.