Let $f:X \to Y$ be a finite map from a normal projective variety to a smooth projective variety, $D$ be a Cartier divisor on $X$. Do we have any relation between $\kappa(X,D)$ and $\kappa(Y,f\_\*D)$?



Kodaira dimensions of push-forward via finite map
There is an obvious relation: the pushforward map $f\_\*:|mD|\rightarrow |f\_\*(mD)|=|mf\_\*D|$ is injective, hence $\kappa (X,D)\leq \kappa (X,f\_\*D)$. It is easy to see that you cannot get more: for instance, take for $f$ a general projection from a cubic surface $X\subset \mathbb{P}^3$ to $\mathbb{P}^2$, and for $D$ a line in $S$. Then $\kappa (S,D)=0$, but $f\_\*D$ is a line in $\mathbb{P}^2$, so $\kappa (\mathbb{P}^2,f\_\*D)= 2$.


