In this paper we discuss topological properties of holomorphic Lefschetzpencils on the four-torus. Relying on the theory of moduli spaces of polarizedabelian surfaces, we first prove that, under some mild assumption, the (smooth)isomorphism class of a holomorphic Lefschetz pencil on the four-torus isuniquely determined by its genus and divisibility. We then explicitly give asystem of vanishing cycles of the genus-3 holomorphic Lefschetz pencil on thefour-torus due to Smith, and obtain those of holomorphic pencils with highergenera by taking finite unbranched coverings. One can also obtain the monodromyfactorization associated with Smith's pencil in a combinatorial way. Thisconstruction allows us to generalize Smith's pencil to higher genera, which isa good source of pencils on the (topological) four-torus. As anotherapplication of the combinatorial construction, for any torus bundle over thetorus with a section we construct a genus-3 Lefschetz pencil whose total spaceis homeomorphic to that of the given bundle.
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