# Fibrations and stable generalized complex structures

## TL;DR

This paper introduces stable generalized complex structures, develops symplectic construction techniques for Lie algebroids, and constructs these structures from log-symplectic and Lefschetz fibrations.

## Contribution

It develops Gompf-Thurston symplectic techniques for Lie algebroids and introduces boundary Lefschetz fibrations to construct stable generalized complex structures.

## Key findings

- Constructed stable generalized complex structures from log-symplectic structures.
- Introduced boundary Lefschetz fibrations for structure construction.
- Connected genus one Lefschetz fibrations over the disk to stable structures.

## Abstract

A generalized complex structure is called stable if its defining anticanonical section vanishes transversally, on a codimension-two submanifold. Alternatively, it is a zero elliptic residue symplectic structure in the elliptic tangent bundle associated to this submanifold. We develop Gompf-Thurston symplectic techniques adapted to Lie algebroids, and use these to construct stable generalized complex structures out of log-symplectic structures. In particular we introduce the notion of a boundary Lefschetz fibration for this purpose and describe how they can be obtained from genus one Lefschetz fibrations over the disk.