Kodaira additivity, birational isotriviality, and specialness

We show that a smooth projective fibration $f\colon X\to Y$ between connected complex quasi-projective manifolds satisfies the equality $\overline{\kappa} (X)=\overline{\kappa}(X_y)+\overline{\kappa} (Y)$ of Logarithmic Kodaira dimensions if its fibres $X_y$ have semi-ample canonical bundles. Without the semi-ampleness assumption, this additivity was conjectured by M. Popa. Several cases are established in a paper by M. Popa and Chr. Schnell which inspired the present text. Although the present results overlap with those of the mentioned paper in the projective case, the approach here is different, based on the rôle played by birationally isotrivial fibrations, special manifolds and the core map of $Y$ introduced and constructed by the author.