Infinitely many Lefschetz pencils on ruled surfaces

TL;DR

The authors address whether ruled surfaces with χ(X) < 0 admit infinitely many inequivalent Lefschetz pencils of fixed genus and base points, constructing such pencils on X and on X # 4\overline{\mathbb{CP}}^2 via partial conjugation of the Matsumoto–Cadavid–Korkmaz fibration. They leverage the Johnson homomorphism and Torelli-group techniques to distinguish fibrations while preserving a fixed fiber class, and then realize these fibrations as symplectic pencils through Gompf–Thurston constructions, showing the regular fibers remain symplectic. The results yield infinitely many pairwise inequivalent Lefschetz fibrations on a fixed smooth 4-manifold and, upon blowing down 4 (-1)-sections, infinitely many inequivalent Lefschetz pencils on ruled surfaces, with all associated fibrations compatible with a common symplectic form. Additionally, the paper demonstrates a broader phenomenon by producing infinitely many inequivalent yet homeomorphic Lefschetz fibrations in a separate setting, contributing to Problem 4.98 in the K3 list and highlighting contrasts between fibrations and fiberings in low-dimensional topology.

Abstract

We show that any ruled surface $X$ with $χ(X) < 0$ admits infinitely many inequivalent Lefschetz pencils of fixed genus and number of base points. Our proof proceeds by building infinitely many inequivalent Lefschetz fibrations on a blow-up $X \# 4 \overline{\mathbb{CP}^2}$ of $X$ with constant fiber class, via a mechanism known as partial conjugation. Furthermore, there exists a symplectic form on $X$ compatible with all such pencils, and similarly for the fibrations in $X\#4\overline{\mathbb{CP}^2}$. This provides the first example of this phenomenon and makes progress on Problem 4.98 of the K3 list of problems in low-dimensional topology in the case of ruled surfaces.

Figures (10)

• Figure 1: Vanishing cycles of the MCK Lefschetz fibration of genus $2g$ and the involution $\eta$.
• Figure 2: Vanishing cycles of the MCK Lefschetz fibration of genus $2g$, the involution $\eta$, and a hyperelliptic involution $\iota$.
• Figure 3: Left: Curves used to define $f \in \mathcal{I}_{2g}$; Right: Lifts $\tilde{x}, \tilde{y}$ of $x, y$ to $\Sigma_{2g}^4$ and their images under $\tilde{h}_i \in \mathop{\mathrm{Mod}}\nolimits(\Sigma_{2g}^4)$ defined in Lemma \ref{['lem:bounding-pair-sections']} for both $i = 1, 2$ (cf. Lemma \ref{['lem:tilde-f-lemma']}). A hyperelliptic involution $\iota$ acting on $\Sigma_{2g}^4$ permuting the boundary components.
• Figure 4: Lifts of the vanishing cycles of the MCK Lefschetz fibration of genus $2g$ to $\Sigma_{2g}^4$ found by Hamada hamada. The four boundary components of $\Sigma_{2g}^4$ are denoted by $\delta_1, \delta_2, \delta_3, \delta_4$. Top left: $B_{0, 1}^1, \dots, B_{2g,1}^1, C_1^1$; Top right: $B_{0, 2}^1, \dots, B_{2g,2}^1, C_2^1$; Bottom left: $B_{0,1}^2, \dots, B_{2g,1}^2, C_1^2$; Bottom right: $B_{0,2}^2, \dots, B_{2g,2}^2, C_2^2$
• Figure 5: Some homology classes in $H$

Theorems & Definitions (105)

• Theorem 1.2
• Theorem 1.3
• Remark 1.4: On the fiber genus
• Remark 1.5: The case of surface bundles over surfaces and mapping tori
• Remark 1.6
• Remark 1.7: Effectiveness of partial conjugations
• Remark 1.8
• Remark 1.9
• Remark 1.10
• Theorem 2.1: Kas kas, Matsumoto matsumoto