This question comes from learning the paper "Existence of minimal models for varieties of log general type", where they define the log terminal model (See Definition 3.6.7). However, the question itself does not need that definition.


Let $\pi: X \to U$ be a projective morphism of normal quasi-projective varieties. Suppose that $K\_X + \Delta$ is log canonical and let $\phi: X -\to Y$ be a birational contraction of normal quasi-projective varieties over $U$, where $Y$ is projective over $U$ (here, $\phi$ is a "birational contraction" means $\phi^{-1}$ does not contract any divisor, for example, when $\phi$ is a morphism). Set $\Gamma=\phi\_\* \Delta$. Then is it true $$\phi\_\*(K\_X + \Delta)=K\_Y +\Gamma\quad?$$


I am equally satisfied with an answer in the simplified case as $U$ is a point, or $\phi$ is a morphism.



Pushforward of a log canonical pair
Yes, since 
$$\phi\_\*K\_X=K\_Y.$$
Here is the reason. We take a common resolution of $X$ and $Y$, say
$$
p:W\rightarrow X, \\q: W \rightarrow Y.
$$
Then we can right 
$$
K\_W-p^\*K\_X=E,\\
K\_W-q^\*K\_Y=F,
$$
where $E$ is exceptional over $X$ (and over $Y$ because "contraction") and $F$ is exceptional over $Y$. If apply $q\_\*$ to 
$$
p^\*K\_X+E=q^\*K\_Y=F,
$$ 
we get the equality.


