Topology of holomorphic Lefschetz pencils on the four-torus

TL;DR

This paper studies the topological properties of holomorphic Lefschetz pencils on the four-torus, providing classification results, explicit vanishing cycles, and methods to generate higher genus pencils and construct specific Lefschetz fibrations.

Contribution

It establishes the uniqueness of holomorphic Lefschetz pencils on the four-torus based on genus and divisibility, and introduces combinatorial methods to construct and generalize these pencils.

Findings

Classification of Lefschetz pencils by genus and divisibility

Explicit vanishing cycles for genus-3 pencil

Construction of higher genus pencils and torus bundle fibrations

Abstract

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.