## Algebraic & Geometric Topology

### Topology of holomorphic Lefschetz pencils on the four-torus

#### 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 3 (2018), 1515-1572.

Dates
Received: 23 January 2017
Revised: 4 September 2017
Accepted: 1 October 2017
First available in Project Euclid: 26 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.agt/1524708099

Digital Object Identifier
doi:10.2140/agt.2018.18.1515

Mathematical Reviews number (MathSciNet)
MR3784012

Zentralblatt MATH identifier
06866406

#### Citation

Hamada, Noriyuki; Hayano, Kenta. Topology of holomorphic Lefschetz pencils on the four-torus. Algebr. Geom. Topol. 18 (2018), no. 3, 1515--1572. doi:10.2140/agt.2018.18.1515. https://projecteuclid.org/euclid.agt/1524708099

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