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Let $k$ be an algebraically closed field of $\mathrm{char}(k)=p>0$, $X$ a smooth toric projective variety of $\dim X=n$, $F_X:X\rightarrow X$ the absolute Frobenius morphism of $X$. Then for any $\mathscr{L}\in Pic(X)$ ${Frob_X}_*\mathscr{L}\cong\bigoplus_{s=1}^{p^{n}}\mathscr{L}_s$, where $\mathscr{L}_s\in Pic(X)$, (See [1], [2]).

My queston is: if $\mathscr{L}$ is anti-ample, then $\mathscr{L}_s$ is anti-ample for any $1\leq s\leq p^n$? I know that if the Picard number $\rho(X)=1$, then the claim is true, but not sure in general case.

[1] Thomsen, J. F. Frobenius direct images of line bundles on toric varieties. J. Algebra 226 (2000), no. 2, 865-874.

[2] Bogvad, R. Splitting of the direct image of sheaves under the Frobenius. Proc. Amer. Math. Soc. 126 (1998), no. 12, 3447-454.

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What happens in $\mathbb{P}^1 \times \mathbb{P}^1$? –  Karl Schwede Apr 30 '11 at 12:10
    
Can you find an example of a toric n-fold $X$ such that for any ample line bundle $L$, $H^0(X,L \otimes \omega_X) \neq 0$? If so, then the top cohomology of any anti-ample line bundle is non-zero, which means that if your question had a positive answer, you would have that $h^n(X,L^{-1})=h^n(X,F_*L^{-1}) \geq p^n$, which cannot happen for $p>>0$. –  Parsa May 1 '11 at 13:26

2 Answers 2

up vote 3 down vote accepted

I think the answer is "no". As I suggested in a now-deleted answer, we can restrict ourselves to considering the restriction to a torus invariant $\mathbb{P}^1$. A line bundle on a toric variety is determined by its restriction to each such $\mathbb{P}^1$, and is anti-ample if and only if its restriction to such a $\mathbb{P}^1$ is.

Let $P$ be a torus invariant $\mathbb{P}^1$. Let the normal bundle of $P$ be $\bigoplus_{i=1}^{n-1} \mathcal{O}(a_i)$. So there is a torus invariant open neighborhood of $P$ which has two affine charts: The first one is $\mathrm{Spec} \ k[t, x_1, \ldots, x_{n-1}]$ and the second is $\mathrm{Spec} \ k[u, y_1, \ldots, y_{n-1}]$ where, on the overlap, we have $t=u^{-1}$ and $y_i = t^{-a_i} x_i$. (I am using that $X$ is smooth. Otherwise, the normal bundle might not be a vector bundle, and I have to work harder.)

Let your line bundle $L$, restricted to $P$, be $\mathcal{O}(-b)$. Since your line bundle is assumed anti-ample, we have $b >0$. Let $q$ and $r$ be generators of your line bundle in the two charts, so $r = t^{-b} q$. (I really hope I'm getting all the signs right. Someone please check!)

We now want to consider the Frobenius pushforward, $F_* L$. This has rank $p^n$. In the first chart, a basis is $ t^{c} \prod x_i^{k_i} q$ for $0 \leq c, \ k_i < p$. In the second chart, a basis is $ u^{d} \prod y_i^{\ell_i} r$ with similar bounds. And $$u^{d} \prod y_i^{\ell_i} r = t^{-b} t^{-d} \prod t^{-a_i \ell_i} x_i^{\ell_i} q = (t^p)^{-N} t^c \prod x_i^{\ell_i} q$$ where $c \equiv -b-d - \sum a_i \ell_i \mod p$, with $0 \leq c < p$, and $N = \left( b+c+d + \sum a_i \ell_i \right)/p$. In other words, $N = \lceil (b+d+\sum a_i \ell_i)/p \rceil$ where $\lceil x \rceil$ is the "round up" function. Our bundle is anti-ample if $N > 0$. The first three terms are positive, but there is plenty of room for the last term to be negative.

In particular, look at the toric surface whose fan has rays in directions $(1,0)$, $(-1, 1)$, $(-1, 0)$, $(-1, -1)$. The four boundary curves have self-intersection, respectively, $(2,0,-2,0)$. Let $L$ be the line bundle whose restrictions to these curves has degree $(-3,-1,-1,-1)$. Let $P$ be the boundary $\mathbb{P}^1$ with self-intersection $-2$, so $a_1=-2$. Then the restriction of $F_* L$ to $P$ has degrees $$\lceil \frac{-1+d - 2 \ell}{p} \rceil$$ where $0 \leq d, \ell < p$.

This is $-1$ whenever $2 (\ell/p) - (d/p) > 1$, which happens for about $1/4$ of the possible values of $(c , \ell)$.

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Thanks very much! I think your answer is right. –  Universe May 3 '11 at 9:07

Let me give another reason for the answer to be no. It probably also gives some insight on when the answer could be yes.

Note the formula $$ F_* \mathcal{O}(D) = \bigoplus_{E\in \mathrm{Pic}\, X} \mathcal{O}(E)^{m(D, E)} $$ where $m(D, E)$ is the number of $T$-divisors $Z = \sum_{\rho\in R} a_\rho D_\rho$ ($R$ denotes the set of rays of the fan, i.e., the prime $T$-divisors) linearly equivalent to $D-pE$ with $0\leq a_\rho < p$.

Let $K$ denote the image of the cube $[0, p)^n$ in $\mathrm{Pic}\, X$. The divisors $E$ with nonzero multiplicity $m(D, E)$ correspond to integer points in $(D - K)/p$. Since $pE$ is (anti-)ample if and only if $E$ is, we see that if $D - K$ is not contained in the (anti-)ample cone then probably such non-(anti-)ample $E$ would exist.

In particular there is an (anti-)ample $B$ (computable given the (anti-)ample cone) with the property that for any (anti-)ample $D$ the line bundles $E$ appearing in $F_* \mathcal{O}(B+D)$ are all (anti-)ample.

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Someone should point out that a (simple) proof of that formula appears in your preprint, front.math.ucdavis.edu/1012.2021 –  Karl Schwede May 2 '11 at 20:13
    
Although I have not read your preprint about Frobenius morphism on toric variety, I think your answer is also right. Thanks very much! –  Universe May 3 '11 at 9:09

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