Fibrations and stable generalized complex structures
TL;DR: In this article, Gompf-Thurston symplectic techniques adapted to Lie algebroids are used to construct stable generalized complex structures out of log-symplectic structures.
read more
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.
read more
Chat with Paper
AI Agents for this Paper
Find similar papers on Google Scholar, PubMed and Arxiv
Write a critical review of this paper
Analyze citations of this paper to find unaddressed research gaps
Citations
•Posted Content
Poisson structures of divisor-type
TL;DR: In this article, a framework for the study of almost-regular Poisson structures of divisor-type and their Lie algebroids is developed, which includes all generically-nondegenerate poisson structures such as log-, $b^k$-, elliptic, elliptic-log, and scattering Poisson structure.
13
Classification of boundary Lefschetz fibrations over the disc
TL;DR: In this paper, it was shown that a four-manifold admits a boundary Lefschetz fibration over the disc if and only if it is diffeomorphic to stable generalized complex structures.
6
Obstructions for Symplectic Lie Algebroids
TL;DR: In this article, the authors discuss topological obstructions to the existence of such Poisson structures, obtained through the characteristic classes of their associated symplectic Lie algebroids.
Fibrations in semitoric and generalized complex geometry
TL;DR: In this article , the interplay between self-crossing boundary Lefschetz fibrations and generalized complex structures was studied, and it was shown that these fibrs arise from the moment maps in semi-toric geometry and use them to construct stable generalized complex four-manifolds using Gompf-Thurston methods for Lie algebroids.
Generalized Luttinger surgery and other cut-and-paste constructions in generalized complex geometry
Biao Xiang
- 31 Jul 2022
TL;DR: In this paper , the authors exploit the affinity between stable generalized complex structures and symplectic structures, and explain how certain constructions coming from symplectic geometry can be performed in the generalized complex setting.
References
•Book
4-manifolds and Kirby calculus
Robert E. Gompf,András I. Stipsicz +1 more
- 01 Jan 1999
TL;DR: In this article, the authors introduce surfaces in 4-manifolds complex surfaces and Kirby calculus, a calculus based on handelbodies and Kirby diagrams, which is used for handel bodies and kirby diagrams.
1.8K
•Posted Content
Generalized complex geometry
TL;DR: In this paper, the concept of a generalized Kahler manifold has been introduced, which is equivalent to a bi-Hermitian geometry with torsion first discovered by physicists.
1.4K
Generalized complex geometry
TL;DR: Generalized complex geometry encompasses complex and symplectic ge- ometry as its extremal special cases as mentioned in this paper, including generalized complex branes, which interpolate be- tween at bundles on Lagrangian submanifolds and holomorphic bundles on complex sub-mansifolds, and the basic properties of this geometry, including its enhanced symmetry group, elliptic deforma- tion theory, relation to Poisson geometry, and local structure theory.
•Book
General theory of lie groupoids and lie algebroids
Kirill C. H. Mackenzie
- 01 Jun 2005
TL;DR: A comprehensive modern account of the theory of Lie groupoids and Lie algebroids, and their importance in differential geometry, in particular their relations with Poisson geometry and general connection theory, is given in this article.
Generalized Calabi-Yau manifolds
TL;DR: A geometrical structure on even-dimensional manifolds is defined in this paper, which generalizes the notion of a Calabi-Yau manifold and also a symplectic manifold.
1.2K