Additivity and Relative Kodaira Dimensions

The notion of Kodaira dimension has been defined for complex manifolds in [16], for symplectic 4−manifolds in [23] (see also [35], [17]). It is shown in [5] (and [23]) that these two definitions are compatible in dimension 4. Furthermore, we calculate it for some (4-dimensional) Lefschetz fibrations when the base has positive genus. In [50], this notion is extended to 3−dimensional manifolds via geometric structures in the sense of Thurston. All these Kodaira dimensions are “absolute” invariants, taking values in the set