### 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 |
---|---|

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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Algebraic and Geometric Topology*,

*18*(3), 1515-1572. https://doi.org/10.2140/agt.2018.18.1515

**Topology of holomorphic lefschetz pencils on the four-torus.** / Hamada, Noriyuki; Hayano, Kenta.

Research output: Contribution to journal › Article

*Algebraic and Geometric Topology*, vol. 18, no. 3, pp. 1515-1572. https://doi.org/10.2140/agt.2018.18.1515

}

TY - JOUR

T1 - Topology of holomorphic lefschetz pencils on the four-torus

AU - Hamada, Noriyuki

AU - Hayano, Kenta

PY - 2018/4/9

Y1 - 2018/4/9

N2 - 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.

AB - 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.

KW - Lefschetz pencil

KW - Mapping class groups

KW - Monodromy factorizations

KW - Polarized abelian surfaces

KW - Symplectic calabiyau four-manifolds

UR - http://www.scopus.com/inward/record.url?scp=85045199430&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85045199430&partnerID=8YFLogxK

U2 - 10.2140/agt.2018.18.1515

DO - 10.2140/agt.2018.18.1515

M3 - Article

AN - SCOPUS:85045199430

VL - 18

SP - 1515

EP - 1572

JO - Algebraic and Geometric Topology

JF - Algebraic and Geometric Topology

SN - 1472-2747

IS - 3

ER -