Kebekus–Kovács conjecture on the Kodaira dimension of bases

Let $X^{\circ}$ be a smooth quasi-projective variety, and let $f^{\circ}:Y^{\circ}\to X^{\circ}$ be a smooth family of canonically polarized varieties with maximal variation. Write $\kappa(X^{\circ})$ for the Kodaira dimension of the base and $\operatorname{Var}(f)$ for the variation of the family. Kebekus–Kovács conjecture. Either

$$\kappa(X^{\circ})=-\infty\quad\text{and}\quad \dim X^{\circ}>\operatorname{Var}(f),$$

or

$$\kappa(X^{\circ})\geq \operatorname{Var}(f).$$

The conjecture generalizes Viehweg's hyperbolicity conjecture to families that are not necessarily of maximal variation. The source records proofs in base dimensions two and three and under suitable minimal-model-program assumptions, and states that Taji proved it in general via Campana's isotriviality conjecture.