The Kodaira Dimension of Lefschetz Fibrations
  In this note, we verify that the complex Kodaira dimension $\kappa^h$ equals the symplectic Kodaira dimension $\kappa^s$ for smooth 4-manifolds with complex and symplectic structures. We also calculate the Kodaira dimension for many Lefschetz fibrations. 
Josef G Dorfmeister and Weiyi Zhang