I need a reference for a proof of the following fact: let $X$ be a toric variety then $X$ is log Fano.


Thanks a lot.



Reference request: log Fano varieties
Here is the detailed argument. First one has to prove the following.


**Lemma 1**
Let $D = \sum\_i d\_iD\_i$ be a $\mathbb{Q}$-divisor on a normal projective variety $X$ such that $d\_i < 1$ and the pair $(X,\lceil D \rceil)$ is lc. Then $(X,D)$ is klt.


*proof)* Let $f:Y\rightarrow X$ be a log resolution of the pair $(X,\lceil D \rceil)$. We have 
$$K\_Y = f^{\*}K\_X+\sum\_ia\_iE\_i$$
and
$${\lceil \widetilde{D} \rceil} = f^\*\lceil D \rceil-\sum\_ib\_i E\_i$$
where $\lceil \widetilde{D} \rceil$ is the strict transform of $\lceil D \rceil$. Therefore, 
$$K\_Y = f^{\*}(K\_X+\lceil D \rceil)+\sum\_i(a\_i-b\_i)-\lceil \widetilde{D} \rceil$$
and since $(X,\lceil D \rceil)$ is lc we have $a\_i-b\_1\geq -1$. On the other hand 
$$\widetilde{D} = f^{\*}D-\sum\_it\_iE\_i$$
with $t\_i<b\_i$ because $d\_i<1$ for any $i$. This yields $a\_i-t\_i > a\_i-b\_i\geq -1$, and the pair $(X,D)$ is klt. 


Now you have:


**Proposition**
Let $X$ be a projective toric variety. Then $X$ is log Fano.


*proof)* Let $D\_1^X,...,D\_r^X$ be the irreducible toric invariant divisors on $X$. Then we have $K\_X = -\sum\_iD\_i^X$, see \cite{Ful}. Now, let $A = \sum\_ia\_iD\_i^X$ be an ample toric invariant divisor, and $\epsilon$ a rational number $0<\epsilon \ll 1$. Therefore
$$-K\_X-\epsilon A = \sum\_i(1-\epsilon a\_i)D\_i^X$$
with $1-\epsilon a\_i < 1$. The divisor $D = \sum\_i(1-\epsilon a\_i)D\_i^X$ is such that $\epsilon A = -K\_X -D$ is ample. Note that $\lceil D\rceil \sim -K\_X$. Let $f:Y\rightarrow X$ be a toric log resolution of $(X,\lceil D\rceil)$, and let $D\_1^Y,...,D\_h^Y$ be the invariant toric divisors on $Y$. We have
$$K\_Y = f^\*(K\_X+\lceil D\rceil)+\sum a\_iE\_i-\lceil \widetilde{D}\rceil = \sum a\_iE\_i-\lceil \widetilde{D}\rceil$$
because $\lceil D\rceil \sim -K\_X$. On the other hand $K\_Y = -\sum\_iD\_i^Y$ yields
$$K\_Y = \sum a\_iE\_i-\lceil \widetilde{D}\rceil = -\sum\_iD\_i^Y.$$
This forces $a\_i = -1$ for any $i$. Therefore, the pair $(X,\lceil D\rceil)$ is lc. To conclude it is enough to apply Lemma 1.


