C^\infty-logarithmic transformations and generalized complex structures

Abstract

Applying logarithmic transformations along 2-tori, we construct a generalized complex structure J_n with n type changing luci for every $n\ge 0$ on genus 1-Lefschetz fibrations with a cusp neighborhood, which include elliptic surfaces with non-zero euler characteristic. Applying a technique of broken Lefschetz fibrations, we further obtain twisted generalized complex structures with arbitrary large numbers of connected components of type changing loci on the manifold which is obtained from a symplectic manifold by logarithmic transformations of multiplicity 0 on a symplectic 2-torus with trivial normal bundle. The connected sums $(2m+1)S^2\times S^2$ for $m\ge 0$, $(2n-1)\C P^2# (10n-1)\ol{\C P^2}$ and $S^1\times S^3$ admit twisted generalized complex structures J_n with n type changing luci for arbitrary large n.

12 pages, 2 figures. Revised version including further examples as an application

References in corpus (1)

• On the number of type change loci of a generalized complex structure

Cited by in corpus (3)

• On the number of type change loci of a generalized complex structure
• Unobstructed deformations of generalized complex structures induced by $C^{\infty }$ logarithmic symplectic structures and logarithmic Poisson structures
• Odd type generalized complex structures on 4-manifolds