Lefschetz fibrations with infinitely many sections

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.