Reference request: log Fano varieties

Question

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.

Answer 1

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.