Noncomplex smooth 4-manifolds with genus-2 Lefschetz fibrations

Abstract

We construct noncomplex smooth 4-manifolds which admit genus-2 Lefschetz fibrations over $S^2$. The fibrations are necessarily hyperelliptic, and the resulting 4-manifolds are not even homotopy equivalent to complex surfaces. Furthermore, these examples show that fiber sums of holomorphic Lefschetz fibrations do not necessarily admit complex structures.