Combinatorial construction of symplectic 6-manifolds via bifibration structures

TL;DR

This work develops a combinatorial framework to study symplectic 6‑manifolds via bifibration structures, linking compatible pairs consisting of a 4‑manifold Lefschetz fibration monodromy with a braid relation to Lefschetz bifibrations over base bundles. It provides a lifting criterion for braids, a constructive theorem guaranteeing Lefschetz bifibrations from compatible data, and a systematic method to compute invariants such as $\chi$, $\pi_1$, and $c_1c_2$ from monodromies; it also gives an almost complex/symplectic realization and proves a concrete Veronese‑inspired example that mirrors lantern‑type relations. The results extend combinatorial methods from 4‑ to 6‑dimensional symplectic geometry, offering a unified approach to generate and analyze 6‑manifolds via pencil‑bifibration structures. A explicit example ties the construction to degree‑2 Veronese embeddings, suggesting deep connections between high‑dimensional fibrations and classical projective geometry. Overall, the paper provides a robust toolkit for building and investigating symplectic 6‑manifolds through monodromy, braid relations, and fibration techniques.

Abstract

A bifibration structure on a $6$-manifold is a map to either the complex projective plane $\mathbb{P}^2$ or a $\mathbb{P}^1$-bundle over $\mathbb{P}^1$, such that its composition with the projection to $\mathbb{P}^1$ is a ($6$-dimensional) Lefschetz fibration/pencil, and its restriction to the preimage of a generic $\mathbb{P}^1$-fiber is also a ($4$-dimensional) Lefschetz fibration/pencil. This object has been studied by Auroux, Katzarkov, Seidel, among others. From a pair consisting of a monodromy representation of a Lefschetz fibration/pencil on a $4$-manifold and a relation in a braid group, which are mutually compatible in an appropriate sense, we construct a bifibration structure on a closed symplectic $6$-manifold, producing the given compatible pair as its monodromies. We further establish methods for computing topological invariants of symplectic $6$-manifolds, including Chern numbers, from compatible pairs. Additionally, we provide an explicit example of a compatible pair, conjectured to correspond to a bifibration structure derived from the degree-$2$ Veronese embedding of the $3$-dimensional complex projective space. This example can be viewed as a higher-dimensional analogue of the lantern relation in the mapping class group of the four-punctured sphere. Our results not only extend the applicability of combinatorial techniques to higher-dimensional symplectic geometry but also offer a unified framework for systematically exploring symplectic $6$-manifolds.

Figures (12)

• Figure 1: Base spaces of bifibrations.
• Figure 2: Simple paths in $D^2$. Note that $\epsilon_i=1,2,3$ when $\mu_i$ is described in green, red, blue, respectively.
• Figure 3: Simple closed curves in $\Sigma_1^8$.
• Figure 4: Simple closed curves in $\Sigma_0^4$.
• Figure 5: Simple closed curves in $\Sigma_0^6$. Each curve is drawn in red (resp. blue) if the index for it in the relation \ref{['eqn:relation single quotient for 5pts']} is $1$ (resp. $2$).

Theorems & Definitions (57)

• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Definition 2.5
• Theorem 2.6
• Theorem 2.7
• Definition 2.8
• Theorem 2.9