Blow-up of generalized complex 4-manifolds

TL;DR

This paper introduces blow-up and blow-down operations for generalized complex 4-manifolds, constructing new structures on certain 4-manifolds and extending the concept of elliptic Lefschetz fibrations to the generalized complex setting.

Contribution

It develops new surgical techniques for generalized complex 4-manifolds and constructs examples on manifolds that lack traditional complex or symplectic structures.

Findings

Constructed generalized complex structures on nCP2 # m ar{CP2} for odd n.

Extended the notion of symplectic elliptic Lefschetz fibrations to generalized complex manifolds.

Demonstrated that certain 4-manifolds admit generalized complex structures without being complex or symplectic.

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.