Let $U$ be a smooth variety, $m >1$ be a positive integer and $D\_{m,U} \in | {-}m K\_U|$ be a smooth irreducible divisor. 
Let $\pi: V\_m:= Spec \bigoplus\_{i=0}^{m-1} 
\mathcal{O}\_U(i K\_U) \rightarrow U$ be a cyclic cover determined by $D\_{m,U}$. 
Assume that $|-K\_U|$ contains a smooth irreducible member $D\_U$ and put $D\_{V\_m}:= \pi\_m^{-1}(D\_U)$.
Consider the logarithmic differential sheaves $\Omega^1\_{V\_m}(\log D\_{V\_m})$ and $\Omega^1\_U(\log D\_U)$. 


**Question 1** $\Omega^1\_{V\_m} (\log D\_{V\_m}) \simeq \pi\_m^\* \Omega^1\_U(\log D\_U)$ ? 


**Question 2** Let $(\pi\_m)\_\* \Omega^1\_{V\_m}(\log D\_{V\_m}) =: \bigoplus \mathcal{F}\_i$ 
be the eigendecompposition with respect to the $\mathbb{Z}/ m\mathbb{Z}$-action where $\mathcal{F}\_i$ is the sheaf of sections of eigenvalue $\zeta^i$ ($\zeta$ is the $m$-th primitive root of unity). 
Is $\mathcal{F}\_1 \simeq \Omega^1\_U (\log D\_U)(-D\_U)$? 
If it's not, can you describe $\mathcal{F}\_i$ explicitly? 


I saw the book by Esnault-Viehweg. Are there other references on that topic? 


(add) I can see that Question 1 is wrong by the arguments in the Sándor Kovács' answer. 



pullback or push forward of logarithmic differential sheaf by cyclic cover 
Question 1 indeed seems right until one reads your comment. It seems right, because
we are thinking that $D\_U$ is the branch divisor. It is indeed right if you take
$D\_U=D\_{m,U}$ in place of your choice.


On the other hand, as stated, Question 1 could not be correct!! Observe that the right
hand side is independent of $m$ while the left hand side can get bigger and bigger
as $m$ grows.


The crux of the matter is that $D\_{V\_m}$ should be compared to $D\_{m,U}$ and not
$D\_U$.


Here is some evidence: Observe that $B=D\_{m,U}$ is the branch divisor of $\pi\_m$. Let
$R=\frac 1m\pi^\*B$ (a $\mathbb Q$-divisor if you want) be the $m^\text{th}$ root of
$B$. Then we have the (logarithmic) ramification formula: $$K\_{V\_m}+ R \sim \pi^\*(K\_U
+ B).\tag{$\*$}$$


Next observe that $B\sim mD\_U$ and hence $D\_U$ cannot contain $B$ (which is assumed
to be irreducible), so $D\_{V\_m}=\pi^\*D\_U$ and then $D\_{V\_m}\sim\_{\mathbb Q}R$. 


So, we obtain by ($\*$) that $$\det\Omega\_{V\_m}(\log D\_{V\_m})\sim\_{\mathbb Q}\det
\Omega\_{V\_m}(\log R)\sim \pi^\*(\det\Omega\_U(\log B)),\tag{+}$$ 
which is strictly bigger than
$\pi^\*(\det\Omega\_U(\log D\_U))$.


**Remarks**


1) It is entirely unimportant that the linear system comes from $-K\_U$. This was not
used anywhere, but in the last sentence one may notice that with that definition
$\det\Omega\_U(\log D\_U)\simeq \mathscr O\_U$, which might make it easier to believe
that the claim and proof here is correct.


2) If one defines $D\_U=D\_{m,U}$, then Question 1 is correct, this is an easy local
calculation.


3) Under #2 above Question 1 implies Question 2. The proof is, as Donu suggests,
simply using the projection formula and then simply applying the same formula for the
push-forward of the structure sheaf for a cyclic cover.


**Addition:**
The recent edit to your question seems to indicate that perhaps you still expect that Question 2 might be true even if Question 1 is not. I don't think that's going to happen either. (I thought this would be kind of clear from the above).


By the above computation and remarks one can see that
$$\pi\_\*\Omega\_{V\_m}(\log R)\sim \Omega\_U(\log B)\otimes \pi\_\*\mathscr O\_{V\_m}\sim \Omega\_U(\log B)\otimes (\oplus\_j\mathscr O\_{U}(-jB)),$$
so the $\zeta$-eigensheaf for this pushforward is 
 $\Omega\_U(\log B)(-B)$.


Now since $\det\Omega\_{V\_m}(\log D\_{V\_m})\sim\_{\mathbb Q}\det
\Omega\_{V\_m}(\log R)$, counting degrees should show you that the answer to Question 2 is also negative, unless $\dim U=1$. On the other hand, if $\dim U=1$, you will be hard pressed to find a $D\_{m,U}$ that is both smooth and irreducible for $m>1$.


