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

Abstract

We show that there are generalized complex structures on all 4manifolds 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 S 2 , (2m -1)CP 2 #lCP 2 and S 1 S 3 admit twisted generalized complex structures J n with n type changing loci for arbitrary large n.