**Theorem 2** [1, p.46] Let $X$ be a non-singular projective algebraic variety of dimension $n$, and $D$ a numerically effective $\mathbb{Q}$-divisor such that $(D^n)>0$. We assume that the support of the non-integral part $D-[D]$ is a divisor with normal crossing on $X$. Then $H^i(X,O\_X([-D]))=0$ for $i<n$.


**Proof.** We write $-D=[-D]+\sum d\_jE\_j$, where $0<d\_j<1$ and $\sum E\_j$ is a divisor with normal crossings. Let $p\_j,q\_j$ be positive integers such that $d\_j=p\_j/q\_j$. Applying Lemma 5 to the $q\_j$ attached to $E\_j$, we obtain non-singular covering $f:X'\to X$ such that $f^\*E\_j=q\_jE\_j'$. Let $f^\*(-D)$ be the divisor on $X'$ defined to be $f^\*([-D])+\sum p\_jE\_j'$. Then $f^\*(-D)$ on $X'$ 
satisfies the condition of Theorem 1. Hence $H^i(X,f\_\*(O\_{X'}(f^\*(-D))))=H^i(X',O\_{X'}(f^\*(-D)))=0$ for $i<n$.


All is Ok for the above. 


On the other hand, the trace map: $f\_\*(O\_{X'})\to O\_X$ extends to a surjective homomorphism $f\_\*O\_{X'}(\sum p\_jE\_j)\to O\_X$ since $d\_j<1$. Hence $O\_X$ is a direct summand of $f\_\*O\_{X'}(\sum p\_jE\_j)$, and $O\_X([-D])$ is that of $f\_\*O\_{X'}(f^\*(-D))$. Therefore, $H^i(X,O\_X([-D]))=0$. Q.E.D.


I am not clear about how the trace map is $f\_\*(O\_{X'})\to O\_X$ is defined and how it extends? And how is this related to $d\_j<1$?
I had thought about the surjective morphism $O\_{\sum q\_jE\_j}\to O\_{\sum p\_jE\_j}\to 0$ 


**Reference**


[1] Yujiro Kawamata, *A Generalization of Kodaira-Ramanujam's Vanishing theorem*, Math. Ann. **262** (1982) pp 43-46 ([link](http://dx.doi.org/10.1007/BF01456407))



A question about kawamata's proof of vanishing for big and nef $\mathbb{Q}$ divisors
ACL already basically answered this in the comments (including what to do in local coordinates), but maybe this is worth explaining in a bit more detail and in more generality.


Whenever one has a finite map between varieties $f : X' \to X$ we have an induced map of fraction fields $K(X) \subseteq K(X')$. Then $K(X')$ is a finite dimensional $K(X)$ vector space and each element $z \in K(X')$ gives a $K(X)$-linear map $\phi\_z : K(X') \xrightarrow{\cdot z} K(X')$. The trace of $z$ is defined to be the trace of the linear map $\phi\_z$. This map is non-zero if and only if $f$ is separable (so in your context, it is definitely nonzero).


This gives us a $K(X)$-linear map $T : K(X') \to K(X)$. Now, it is an easy exercise (in Atiyah-Macdonald) that $T$ sends $f\_\* O\_{X'}$ to $O\_X$ as long as $X$ is normal. From now on, let's assume that both $X$ and $X'$ are normal (which is your situation as well). It is also easy to see that $T(f\_\* O\_{X'}) = O\_X$ since $T(1)$ is just the degree of the map mod the characteristic (which is not zero in characteristic zero). 


More generally though, if one chooses $K\_X$ to be a canonical divisor on $X$ and sets $K\_{X'} = f^\* K\_X + \text{Ram}$ where $\text{Ram}$ is the ramification divisor, then $T$ also sends $f\_\* O\_{X'}(K\_{X'})$ to $O\_X(K\_X)$ and it's easy to see that this is equivalent to sending $f\_\* O\_{X'}(\text{Ram})$ to $O\_X$ (if $K\_X$ is Cartier, as in your case, then it's trivial, if $K\_X$ is not Cartier, it is Cartier on the smooth locus and then you can reflexify elsewhere). 


Ok, so how does this help you?


Note $X' \to X$ is a cyclic cover, and its ramified over $D$ and the ramification divisor is probably exactly $\sum (q\_j-1) E\_j'$ since the ramification is tame (we are in characteristic zero) -- I didn't look up the appropriate lemmas in Kawamata's paper, but this is how it should work. Regardless, the point is that $\sum p\_j E\_j$ is less than the ramification divisor and so we have


$$
f\_\* O\_{X'}( \sum p\_j E\_j) \hookrightarrow f\_\* O\_{X'}(\text{Ram}) \xrightarrow{T} O\_X
$$


which I think is exactly the map you want.


For references for this, I think it is nearly all in Serre's Local Fields among other places. But if you need more references, I can track down some information.


### Final comment


It is probably worth observing that in fact $f\_\* O\_{X'}(\text{Ram})$ is the largest $O\_X$-module subsheaf of $f\_\* K(X')$ that you can restrict trace to such that the image is still contained $O\_X$.


