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# Journal of Symplectic Geometry

## Volume 14 (2016)

### Number 2

### $C^{\infty}$-logarithmic transformations and generalized complex structures

Pages: 341 – 357

DOI: http://dx.doi.org/10.4310/JSG.2016.v14.n2.a1

#### Authors

#### Abstract

We show that there are generalized complex structures on all 4-manifolds obtained by logarithmic transformations with arbitrary multiplicity along symplectic tori with trivial normal bundle. Applying a technique of broken Lefschetz fibrations, we obtain generalized complex structures with arbitrary large numbers of connected components of type changing loci on every manifold which is obtained from a symplectic 4-manifold by a logarithmic transformation of multiplicity $0$ along a symplectic torus with trivial normal bundle. Elliptic surfaces with non-zero euler characteristic and the connected sums $(2m - 1) S^2 \times S^2, (2m - 1) \mathbb{C}P^2 \# l \overline{\mathbb{C}P^2}$ and $S^1 \times S^3$ admit twisted generalized complex structures $\mathcal{J}_n$ with $n$ type changing loci for arbitrary large $n$.