TY - JOUR

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

AU - Hamada, Noriyuki

AU - Hayano, Kenta

N1 - Funding Information:
Acknowledgements The authors would like to thank Refik ˙nanç Baykur and Tian-Jun Li for fruitful discussions and helpful comments on an earlier draft of this paper. Hayano was supported by JSPS KAKENHI (26800027 and 17K14194).

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

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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 -