2024/09/23 by Seraphina Eun Bi Lee, Lee, Seraphina Eun Bi, Carlos A. Serván +1 · 1 citation
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Holomorphic and Operator Theory #Point processes and geometric inequalities #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2409.15265
openalex publication_date 2024/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Arakelov--Parshin rigidity theorem implies that a holomorphic Lefschetz fibration π: M → S2 of genus g ≥ 2 admits only finitely many holomorphic sections σ:S2 → 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.