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Topology of holomorphic Lefschetz pencils on the four-torus

Noriyuki Hamada and Kenta Hayano

Algebraic & Geometric Topology 18 (2018) 1515–1572

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.

Lefschetz pencil, polarized abelian surfaces, symplectic Calabi–Yau four-manifolds, monodromy factorizations, mapping class groups
Mathematical Subject Classification 2010
Primary: 57R35
Secondary: 14D05, 20F38, 32Q55, 57R17
Received: 23 January 2017
Revised: 4 September 2017
Accepted: 1 October 2017
Published: 3 April 2018
Noriyuki Hamada
Department of Mathematics and Statistics
University of Massachusetts
Amherst, MA
United States
Kenta Hayano
Department of Mathematics
Keio University
Yagami Campus