Algebraic & Geometric Topology

Topology of holomorphic Lefschetz pencils on the four-torus

Noriyuki Hamada and Kenta Hayano

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R35: Differentiable mappings
Secondary: 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.) 20F38: Other groups related to topology or analysis 32Q55: Topological aspects of complex manifolds 57R17: Symplectic and contact topology

Lefschetz pencil polarized abelian surfaces symplectic Calabi–Yau four-manifolds monodromy factorizations mapping class groups


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.

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