Lefschetz fibrations with infinitely many sections

TL;DR

This paper demonstrates that unlike holomorphic Lefschetz fibrations, smooth and symplectic Lefschetz fibrations can have infinitely many sections, challenging existing finiteness theorems and providing new examples and criteria.

Contribution

It establishes a criterion for symplectic Lefschetz fibrations to have infinitely many sections and presents examples where finiteness does not hold.

Findings

Holomorphic Lefschetz fibrations have finitely many sections due to Arakelov--Parshin theorem.

Smooth and symplectic Lefschetz fibrations can have infinitely many sections.

Finiteness of sections is not guaranteed when considering automorphisms.

Abstract

The Arakelov--Parshin rigidity theorem implies that a holomorphic Lefschetz fibration $\pi: M \to S^2$ of genus $g \geq 2$ admits only finitely many holomorphic sections $\sigma:S^2 \to M$. We show that an analogous finiteness theorem does not hold for smooth or for symplectic Lefschetz fibrations. We prove a general criterion for a symplectic Lefschetz fibration to admit infinitely many homologically distinct sections and give many examples satisfying such assumptions. Furthermore, we provide examples that show that finiteness is not necessarily recovered by considering a coarser count of sections up to the action of the (smooth) automorphism group of a Lefschetz fibration.