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2018 Topology of holomorphic Lefschetz pencils on the four-torus
Noriyuki Hamada, Kenta Hayano
Algebr. Geom. Topol. 18(3): 1515-1572 (2018). DOI: 10.2140/agt.2018.18.1515

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.

Citation

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Noriyuki Hamada. Kenta Hayano. "Topology of holomorphic Lefschetz pencils on the four-torus." Algebr. Geom. Topol. 18 (3) 1515 - 1572, 2018. https://doi.org/10.2140/agt.2018.18.1515

Information

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

zbMATH: 06866406
MathSciNet: MR3784012
Digital Object Identifier: 10.2140/agt.2018.18.1515

Subjects:
Primary: 57R35
Secondary: 14D05, 20F38, 32Q55, 57R17

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.18 • No. 3 • 2018
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