Blow-up of generalized complex 4-manifolds

Abstract

We introduce blow-up and blow-down operations for generalized complex 4-manifolds. Combining these with a surgery analogous to the logarithmic transform, we then construct generalized complex structures on nCP2 # m \bar{CP2} for n odd, a family of 4-manifolds which admit neither complex nor symplectic structures unless n=1. We also extend the notion of a symplectic elliptic Lefschetz fibration, so that it expresses a generalized complex 4-manifold as a fibration over a two-dimensional manifold with boundary.

25 pages, 15 figures. This is the final version, which was published in J. Top

References in corpus (4)

• Generalized complex geometry
• A surgery for generalized complex structures on 4-manifolds
• Generalized complex structures and Lie brackets
• Generalized complex geometry

Cited by in corpus (18)

• Stable generalized complex structures
• On the number of type change loci of a generalized complex structure
• Blowing up generalized Kahler 4-manifolds
• Poisson modules and generalized geometry
• Semichiral Fields on S^2 and Generalized Kähler Geometry
• Constructions of generalized complex structures in dimension four
• Local classification of generalized complex structures
• Unobstructed deformations of generalized complex structures induced by $C^{\infty }$ logarithmic symplectic structures and logarithmic Poisson structures
• On the local and global classification of generalized complex structures
• Fibrations and stable generalized complex structures
• Complex Dirac structures: invariants and local structure
• C^\infty-logarithmic transformations and generalized complex structures
• Fibrations in semi-toric and generalized complex geometry
• Lagrangian branes with boundary and symplectic methods for stable generalized complex manifolds
• Blow-up formulae for twisted cohomologies with supports
• Odd type generalized complex structures on 4-manifolds
• Generalized Luttinger surgery and other cut-and-paste constructions in generalized complex geometry
• From generalized Kaehler to generalized Sasakian structures