## Abstract

We discuss topological properties of holomorphic Lefschetz pencils on the four-torus. Relying on the theory of moduli spaces of polarized abelian surfaces, we first prove that, under some mild assumptions, the (smooth) isomorphism class of a holomorphic Lefschetz pencil on the four-torus is uniquely determined by its genus and divisibility. We then explicitly give a system of vanishing cycles of the genus-3 holomorphic Lefschetz pencil on the four-torus due to Smith, and obtain those of holomorphic pencils with higher genera by taking finite unbranched coverings. One can also obtain the monodromy factorization associated with Smith’s pencil in a combinatorial way. This construction allows us to generalize Smith’s pencil to higher genera, which is a good source of pencils on the (topological) four-torus. As another application of the combinatorial construction, for any torus bundle over the torus with a section we construct a genus-3 Lefschetz pencil whose total space is homeomorphic to that of the given bundle.

Original language | English |
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Pages (from-to) | 1515-1572 |

Number of pages | 58 |

Journal | Algebraic and Geometric Topology |

Volume | 18 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2018 Apr 9 |

## Keywords

- Lefschetz pencil
- Mapping class groups
- Monodromy factorizations
- Polarized abelian surfaces
- Symplectic calabiyau four-manifolds

## ASJC Scopus subject areas

- Geometry and Topology