Abstract: M. Popa conjectured that if $f: X\to Y$ is a projective and submersive map beween complex quasi-projective manifolds, then $\overline{\kappa}(X)=\kappa(X_y)+\overline{\kappa}(Y)$, where $\overline{\kappa}$ is the Logarithmic Kodaira dimension. We prove this, assuming that the fibres $X_y$ have good minimal models.

Bio-Sketch: Professor Frédéric Campana is a professor emeritus at the Institut Élie Cartan de Lorraine, specializing in complex and algebraic geometry. He has made significant contributions to the study of rationally connected varieties, Kähler geometry, and hyperbolicity in algebraic geometry. Professor Campana has been particularly influential in developing the theory of special and general type varieties. His research often explores the deep interplay between arithmetic and geometric properties of algebraic varieties. He has been an honorary member of the Institut Universitaire de France (IUF) since 2011. In recognition of his contributions, he was awarded the Prix Petit-Dormoyand the Prix Jaffé by the French Academy of Sciences (Académie des Sciences) in 1995 and 2008 respectively.