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arXiv · 2610.01776

Sections, pseudosections, and multisections of Lefschetz fibrations

R. Inanc Baykur·Noriyuki Hamada
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Abstract

We show that not every symplectic Lefschetz fibration over the $2$--sphere admits a smooth section, settling a long-standing open problem. To prove this, we develop a method for studying sections via lifts of monodromy factorizations with point-pushing maps and the induced handle decomposition of the total space, and apply it to an infinite family of Lefschetz fibrations. In contrast, we show that every Lefschetz fibration over $S^2$ admits positive multisections. Some of these examples admit no multisections with spherical components. We also construct signature-zero Lefschetz fibrations of every odd genus $g\geq 7$, which in turn give symplectic Lefschetz fibrations of any prescribed signature in each of these genera.

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R. Inanc Baykur, Noriyuki Hamada. 2026-10-01. Sections, pseudosections, and multisections of Lefschetz fibrations. https://arxiv.org/abs/2610.01776

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