Fibrations in semi-toric and generalized complex geometry

TL;DR

This paper explores the relationship between boundary Lefschetz fibrations and generalized complex structures, demonstrating their origin from semi-toric moment maps and constructing new four-manifolds with these properties.

Contribution

It introduces a method to construct self-crossing stable generalized complex four-manifolds from fibrations, expanding understanding of their structure and singularity interactions.

Findings

Fibrations arise from semi-toric moment maps

Construction of new generalized complex four-manifolds

Compatibility of fibrations with connected sums

Abstract

This paper studies the interplay between self-crossing boundary Lefschetz fibrations and generalized complex structures. We show that these fibrations arise from the moment maps in semi-toric geometry and use them to construct self-crossing stable generalized complex four-manifolds using Gompf--Thurston methods for Lie algebroids. These results bring forth further structure on several previously known examples of generalized complex manifolds. We moreover show that these fibrations are compatible with taking connected sums, and use this to prove a singularity trade result between two types of singularities occurring in these fibrations.