Combinatorial construction of symplectic 6-manifolds via bifibration structures

TL;DR

This paper introduces a combinatorial method to construct symplectic 6-manifolds using bifibration structures, linking monodromy data with topological invariants, and provides explicit examples including a higher-dimensional analogue of the lantern relation.

Contribution

It develops a new framework for constructing symplectic 6-manifolds from compatible monodromy pairs and braid relations, extending combinatorial techniques to higher dimensions.

Findings

Constructed bifibration structures from monodromy and braid relations.

Established methods for computing topological invariants like Chern numbers.

Provided an explicit example related to the degree-2 Veronese embedding.

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.