Veech fibrations

TL;DR

This paper studies complex surfaces fibered over Teichmüller curves with Veech surfaces as fibers, analyzing their invariants and monodromy actions, especially for algebraically primitive cases.

Contribution

It provides explicit computations of topological and complex invariants for Veech fibrations, extending understanding of their geometric structure.

Findings

Computed invariants for all known algebraically primitive Teichmüller curves.

Established the relationship between monodromy action and surface invariants.

Identified the nature of these surfaces as elliptic or of general type depending on fiber genus.

Abstract

We investigate complex surfaces that fiber over Teichm\"uller curves where the generic fiber is a Veech surface. When the fiber has genus one, these surfaces are elliptic fibrations; for higher genus fibers, they are typically minimal surfaces of general type. We compute the topological and complex-geometric invariants of these surfaces via the monodromy action on the mod-$m$ homology of the fiber. We get exact values of the invariants for all known algebraically primitive Teichm\"uller curves.