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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.
Keywords: Lefschetz pencil, polarized abelian surfaces, symplectic Calabi–Yau four-manifolds, monodromy factorizations, mapping class groups
Hamada, Noriyuki 1 ; Hayano, Kenta 2
@article{10_2140_agt_2018_18_1515,
author = {Hamada, Noriyuki and Hayano, Kenta},
title = {Topology of holomorphic {Lefschetz} pencils on the four-torus},
journal = {Algebraic and Geometric Topology},
pages = {1515--1572},
publisher = {mathdoc},
volume = {18},
number = {3},
year = {2018},
doi = {10.2140/agt.2018.18.1515},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.1515/}
}
TY - JOUR AU - Hamada, Noriyuki AU - Hayano, Kenta TI - Topology of holomorphic Lefschetz pencils on the four-torus JO - Algebraic and Geometric Topology PY - 2018 SP - 1515 EP - 1572 VL - 18 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.1515/ DO - 10.2140/agt.2018.18.1515 ID - 10_2140_agt_2018_18_1515 ER -
%0 Journal Article %A Hamada, Noriyuki %A Hayano, Kenta %T Topology of holomorphic Lefschetz pencils on the four-torus %J Algebraic and Geometric Topology %D 2018 %P 1515-1572 %V 18 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.1515/ %R 10.2140/agt.2018.18.1515 %F 10_2140_agt_2018_18_1515
Hamada, Noriyuki; Hayano, Kenta. Topology of holomorphic Lefschetz pencils on the four-torus. Algebraic and Geometric Topology, Tome 18 (2018) no. 3, pp. 1515-1572. doi: 10.2140/agt.2018.18.1515
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