Bi-algebraic geometry of strata of differentials in genus zero

Abstract

We study algebraic subvarieties of strata of differentials in genus zero satisfying algebraic relations among periods. The main results are Ax-Schanuel and Andr\'e-Oort-type theorems in genus zero. As a consequence, one obtains several equivalent characterizations of bi-algebraic varieties. It follows that bi-algebraic varieties in genus zero are foliated by affine-linear varieties. Furthermore, bi-algebraic varieties with constant residues in strata with only simple poles are affine-linear. Additionally, we produce infinitely many new linear varieties in strata of genus zero, including infinitely many new examples of meromorphic Teichm\"uller curves.