Hodge–Teichmüller planes and finiteness results for Teichmüller curves

Abstract

We prove that there are only finitely many algebraically primitive Teichmüller curves in the minimal stratum in each prime genus at least 3. The proof is based on the study of certain special planes in the first cohomology of a translation surface which we call Hodge–Teichmüller planes. We also show that algebraically primitive Teichmüller curves are not dense in any connected component of any stratum in genus at least 3; the closure of the union of all such curves (in a fixed stratum) is equal to a finite union of affine invariant submanifolds with unlikely properties. Results of this type hold even without the assumption of algebraic primitivity. Combined with work of Nguyen and the second author, a corollary of our results is that there are at most finitely many nonarithmetic Teichmüller curves in $\mathcal{H}(4{)}^{\mathrm{hyp}}$.