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Veech fibrations

Sam Freedman, Trent Lucas

TL;DR

We introduce Veech fibrations, Lefschetz fibrations over Teichmüller curves whose smooth fibers are Veech surfaces, and develop a framework to compute their topological and complex-geometric invariants via monodromy on mod- m homology. The core method hinges on the holonomy map and a key criterion ensuring the mod- p image of Aff^+(X,ω) is large (SL(2, F_{p^g})) for infinitely many primes in algebraically primitive cases, enabling explicit invariant formulas for congruence Veech fibrations. Applying this to all known algebraically primitive families (Weierstrass eigenforms in genus 2, regular polygons, and sporadic E7/E8), we produce exact Euler characteristics, signatures, base genera, and BMY-status, with many examples being minimal of general type and, in genus-1 fiber cases, elliptic modular surfaces; notable instances include simply connected Horikawa surfaces such as the double pentagon fibration. The results illuminate the rich 4-manifold topology arising from translation surfaces and provide a concrete path to further exploration of topology, geometry, and arithmetic of Veech fibrations via Thurston–Veech construction and congruence covers.

Abstract

We investigate complex surfaces that fiber over Teichmüller 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 Teichmüller curves.

Veech fibrations

TL;DR

We introduce Veech fibrations, Lefschetz fibrations over Teichmüller curves whose smooth fibers are Veech surfaces, and develop a framework to compute their topological and complex-geometric invariants via monodromy on mod- m homology. The core method hinges on the holonomy map and a key criterion ensuring the mod- p image of Aff^+(X,ω) is large (SL(2, F_{p^g})) for infinitely many primes in algebraically primitive cases, enabling explicit invariant formulas for congruence Veech fibrations. Applying this to all known algebraically primitive families (Weierstrass eigenforms in genus 2, regular polygons, and sporadic E7/E8), we produce exact Euler characteristics, signatures, base genera, and BMY-status, with many examples being minimal of general type and, in genus-1 fiber cases, elliptic modular surfaces; notable instances include simply connected Horikawa surfaces such as the double pentagon fibration. The results illuminate the rich 4-manifold topology arising from translation surfaces and provide a concrete path to further exploration of topology, geometry, and arithmetic of Veech fibrations via Thurston–Veech construction and congruence covers.

Abstract

We investigate complex surfaces that fiber over Teichmüller 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- homology of the fiber. We get exact values of the invariants for all known algebraically primitive Teichmüller curves.
Paper Structure (66 sections, 28 theorems, 115 equations, 6 figures, 4 tables)

This paper contains 66 sections, 28 theorems, 115 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Main problem and results
  3. About the proofs
  4. Connection to elliptic fibrations
  5. Simply connected Veech fibrations
  6. Previous work
  7. Contents
  8. Acknowledgments
  9. Background
  10. Veech surfaces
  11. Veech surfaces
  12. Cylinder decompositions and parabolic elements
  13. Teichmüller curves
  14. Action on homology of Veech surfaces
  15. Veech fibrations
  16. ...and 51 more sections

Key Result

Theorem 1.1

Let $(X,\omega)$ be the regular $q$-gon surface of genus $g = (q - 1)/2$ where $q \ge 5$ is prime. Fix a prime $p \geq 3$ for which the minimal polynomial of $4\cos\left(\pi/q\right)^2$ is irreducible over $\mathbb{F}_p$, and let $\widetilde{\mathbb{X}}_{q,p} \rightarrow \overline{B}_{q,p}$ be the $

Figures (6)

  • Figure 4.1: The Eierlegende Wollmilschau. Side identifcations are given by numbers; unlabeled sides are identified with their opposite.
  • Figure 6.1: The L-shaped polygon generating the Weierstrass curves in genus 2. We use the core curves $\gamma$ and $\delta$ in verifying the hypotheses of \ref{['thm:action_on_homology_mod_p']}
  • Figure 6.2: Chern numbers $(c_2, c_1^2)$ for Veech fibrations $\mathbb{X}_{D, p}$ with $p = 5, 7, 11$. We only plot discriminants $D$ for which $D$ is a nonresidue. Note that the scales of the axes use scientific notation; the relevant powers of 10 are at the upper left and lower right corners.
  • Figure 6.3: Examples of the surface $(X_n,\omega_n)$
  • Figure 6.4: The inductive step of \ref{['lem:staircase-heights']}
  • ...and 1 more figures

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Proposition 2
  • Remark 1
  • proof
  • ...and 48 more