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Non-existence of Shimura curves of Mumford type generically in the non-hyperelliptic locus

Xin Lu, Shengli Tan, Kang Zuo

Abstract

We show that there does not exist any Shimura curve with strictly maximal Higgs field generically in the Torelli locus of non-hyperelliptic curves of genus $g\geq 4$. In particular, Shimura curves of Mumford type are not generically in the Torelli locus of non-hyperelliptic curves of genus $g\geq 4$.

Non-existence of Shimura curves of Mumford type generically in the non-hyperelliptic locus

Abstract

We show that there does not exist any Shimura curve with strictly maximal Higgs field generically in the Torelli locus of non-hyperelliptic curves of genus . In particular, Shimura curves of Mumford type are not generically in the Torelli locus of non-hyperelliptic curves of genus .
Paper Structure (7 sections, 7 theorems, 59 equations)

This paper contains 7 sections, 7 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Shimura curves of Mumford type
  4. The Higgs bundle associated to a Shimura curve
  5. The family of semi-stable curves representing a Shimura curve generically in the Torelli locus
  6. Slope inequality
  7. The proof of \ref{['thm-main-2']}

Key Result

Theorem 1.1

There does not exist any Shimura curve of Mumford type contained generically in the Torelli locus of non-hyperelliptic curves of genus $g=4$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Conjecture 1: Coleman-Oort conjecture
  • Theorem 1.2
  • Corollary 1
  • Definition 1: vz-03
  • Theorem 2.1: moller-11vz-04
  • Definition 2
  • Definition 3
  • Proposition 1: lu-zuo-19-2
  • Corollary 2
  • ...and 7 more