Fibrations and stable generalized complex structures

Gil R Cavalcanti,Ralph L Klaasse

doi:10.1112/plms.12199

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.

Highlights

A generalized complex structure is called stable if its defining anticanonical section vanishes transversally, on a codimension-two submanifold
In [8] it was noted that stable generalized complex structures can alternatively be viewed as symplectic forms on a Lie algebroid, the elliptic tangent bundle, constructed out of the anticanonical section and its zero set
We introduce Lie algebroid fibrations and Lie algebroid Lefschetz fibrations, and give criteria when these can be equipped with compatible Lie algebroid symplectic structures

Summary

Stable generalized complex and log-symplectic structures

We recall the notion of a stable generalized complex structure as defined in [8], and that of log-symplectic structures [17]. The anticanonical bundle KJ∗ of a stable generalized complex manifold together with its natural section s are a particular example of a divisor, which we will introduce shortly. A divisor on X is a pair (U, σ) where U → X is a real/complex line bundle and σ ∈ Γ(U ) is a section whose zero set is nowhere dense. By definition the anticanonical bundle KJ∗ of a stable generalized complex structure together with its natural section s is an example of a complex log divisor. Stable generalized complex structures J on X are in one-to-one correspondence with certain types of elliptic Poisson structures π via the map J → πJ (see Theorem 3.31). When no elliptic divisor structure is specified on D, we say that (X, D) admits a stable generalized complex structure if there exists some elliptic divisor structure |D| on D such that (X, |D|) admits a stable generalized complex structure

Lie algebroids and Lie algebroid symplectic structures

Constructing Lie algebroid symplectic structures

Boundary maps and boundary Lefschetz fibrations

Constructing boundary Lefschetz fibrations

Constructing stable generalized complex structures

Examples and applications