Let $W\_n$ be a set of a log pair having the following property:


For any $(X, D) \in W\_n$


(1)$X$ has dimensional $n$ with tirvial canonical divisor (i.e.$K\_X = 0$). Moreover, $X$ is a $\mathbb{Q}$-factorial variety with canonical singularities. (one can also assume $(X, \frac{1}{2} D)$ is klt).


(2) $D$ is an **integral Weil divisor** which is **big**.


Is it true that there exists a universal $\delta > 0$ (i.e. only depends on $n$), such that we can decompose $D$ in $\mathbb{Q}$-linearly equivalent as follows:
$$D \sim\_\mathbb{Q} A + E,$$ where $A$ is a nef and big divisor with **coefficients** bigger than $ \delta$, and $E$ is an effective divisor?


Any suggestions related to the topic are welcome!!



Decompose a big divisor as nef big divisor and effective divisor
I think the answer is (almost, i.e. up to birational modification) yes but non-trivial see Theorem 1.3 of arXiv:1208.4150 "ACC for log canonical thresholds" by Christopher Hacon, James McKernan, Chenyang Xu.
If you assume that $D$ is integral and $(X,D/2)$ is klt, then since $K\_X+D/2=D/2$ is big, the above theorem says that there exists a uniform positive integer (depending only on the dimension of $X$) such that $|m(K\_X+D/2)|$ defines a birational map $\phi :X\to \mathbb P ^N=|m(K\_X+D/2)|=|(m/2)D|$. If $\phi$ is a morphism (which can be arranged by replacing $X$ by a higher model), then $(m/2)D=\phi ^\* \mathcal O \_{\mathbb P ^N}(1)+E$ where $E$ is an effective divisor and of course $\phi ^\* \mathcal O \_{\mathbb P ^N}(1)$ is nef and big.


I am not sure (and skeptical about) how to do this without replacing $X$ by a higher model. 


