  In this paper 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 assumption, 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.
