${{\mathcal{F}}}\ne 0$ be a $IT^0$ sheaf on an abelian variety $A$, i.e. for every $i > 0$ we have $H^i(A, {{\mathcal{F}}}{{\otimes}}P) =0 $ for all $P \in
{\text{\rm Pic}}^0(A)$. Then $H^0(A,{{\mathcal{F}}}) \ne 0$.

The proof can be found in [@CH], but we reprove it here.

Suppose on the contrary that $H^0(A, {{\mathcal{F}}}) =0$. Since ${{\mathcal{F}}}$ is $IT^0$, the Fourier-Mukai transform of ${{\mathcal{F}}}$ is a locally free sheaf of rank $= h^0(A, {{\mathcal{F}}})$, hence the zero sheaf. The only sheaf that transforms to the zero sheaf is the zero sheaf, which is a contradiction.

Let $f: X' \to X$ be an embedded resolution for $(X,L)$. It is clear that $f^*L$ is nef and big with simple normal crossing support. Let $\Delta:=\lceil f^*L \rceil-f^*L$, then $(X', \Delta)$ is Kawamata log terminal (klt for short; for its definition and property, see [@KMM], Def 0-2-10). By a property of nef and big divisor (see e.g. [@La], ex 2.2.17), there is an effective divisor $N$ such that $A_k:=f^*L-\frac{1}{k} N $ is ample for all $k  \gg 0$. We fix $k$ such that $(X', \Delta+ \frac{1}{k} N)$ is klt. Now we can write $A_k=
({\text{\rm alb}}\circ f)^*M +E$ for some ample $\bf{Q}$-divisor $M$ on $A:={\text{\rm Alb}}(X)$ 