Lefschetz fibrations on the Milnor fibers of cusp and simple elliptic singularities

TL;DR

This work constructs an explicit $S^1$-family of genus-one Lefschetz fibrations over $D^2$ on the Milnor fibers of cusp and simple elliptic singularities in ${\mathbb C}^3$, realized as Lagrangian torus fibrations. It connects these fibrations to Lawson-type foliations on $S^5$ by showing they arise as pullbacks of the Reeb foliation on $S^3$, and derives leafwise symplectic structures via foliated Lefschetz fibrations. A key contribution is the smooth decomposition of K3 surfaces by gluing Milnor fibers corresponding to extended strange duality pairs, yielding a diffeomorphism to K3 and revealing a deep bridge between singularity theory, 4-manifold topology, and lattice dualities. The paper also develops a comprehensive framework for Milnor lattices and monodromy within the Lefschetz-fibration picture, and extends the results to the Inose fibration and related Hilbert modular cusp phenomena. Overall, it provides new topological and geometric tools linking singularities, Lefschetz fibrations, K3 decompositions, and foliated symplectic structures with potential implications for both pure geometry and foliation theory.

Abstract

We show that the total space of the Milnor fibration associated with any cusp or simple elliptic singularity in complex three variables admits an $S^1$-parametric genus-one Lefschetz fibration structure over the $2$-disk. As a consequence, we demonstrate that the Lawson type foliations on $S^5$ associated with such singularities can be regarded as the pullback of the Reeb foliation on $S^3$. This enables us to provide an alternative proof of a previous result by the third author, which states that every Lawson type foliation admits a leafwise symplectic structure. Also we see that a pair of such Milnor fibers can be glued together along boundary into a closed oriented 4-manifold exactly when the pair corresponds to one of the ten extended strange duality pairs among the cusp singularities. This gluing is compatible with the Lefschetz fibrations and the resultant 4-manifold is diffeomrphic to a K3 surface.

Figures (8)

• Figure 1: The left-hand shows the $p+q+r-1$ spheres. The singular fibers are the unions of $p(=2)$, $q(=3)$, and $r(=7)$ spheres which look like rosaries. The dotted spheres $\Sigma_{m,0}$ ($m=1,2,3$) are removed. Instead, the three disks $D_m$ over the bolded segments tie the broken rosaries to the regular fiber $T^2$ at the triangular region $T_+$ (or $T_-$). The right-hand shows how to cut $T_\pm$ out of $T^2$. Here the front triangle is the positive region $T_+$ which are seen from the back.
• Figure 2: The critical values of $g|_Y$ in the left-hand go into pieces in the right-hand. We suppose that the critical points over $P_j$, $Q_k$, $R_l$ satisfy $\arg x=\frac{2\pi j}{p}$, $\arg y=\frac{2\pi k}{q}$, $\arg z=\frac{2\pi l}{r}$ ($j\in {\mathbb Z}_p$, $k\in{\mathbb Z}_q$, $l\in {\mathbb Z}_r$).
• Figure 3: The image of the sphere $\Sigma_+$ is changed from the union of the bold line segments to that of the dotted arcs. Each of the numbers indicates the sign of the intersection over there.
• Figure 4: The Milnor lattice. The left-hand diagram $\widetilde{\mathcal{T}}(p,q,r)$ is equivalent to the right-hand one $\widetilde{\mathcal{T}}'(p,q,r)= {\mathcal{T}}(p,q,r) \oplus \langle[T^{2}]\rangle$.
• Figure 5: The Dynkin diagram $\mathcal{T}(p,q,r)$

Theorems & Definitions (69)

• Definition 1.1: simple elliptic singularity
• Theorem 1.2: Saito S
• Definition 1.3: cusp singularity
• Theorem 1.4: Karras Kar
• Theorem 1.5: Neumann
• Definition 1.6: Milnor radius, singularity link
• Theorem 1.7: Milnor M
• Definition 1.8: Milnor fibration, Milnor fiber, Milnor tube
• Lemma 1.9
• proof