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-
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-
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
