Let $(X,T)$ be a smooth complex toric variety of dimension $d$ with torus $T$ and toric boundary $D=X\setminus T$. Let $\phi : X\to X$ be a finite endomorphism of $X$ such that the restriction


$$\phi|\_{T}:T\to T$$


is the power map $t\mapsto t^{\ell}$.


Let $\mathscr{L}$ be a line bundle on $X$. Then its pushforward $\phi\_\*\mathscr{L}$ will still be locally free, and will have rank $\delta=\ell^d$.


Is it possible to describe such pushforward? For example, is it reasonable to expect something like $\phi\_\*\mathscr{L}\simeq \mathscr{L}^{\oplus\delta}\otimes\mathcal{O}(D)$?


What about the case $X=\mathbb{P}^d$, $\mathscr{L}=\mathcal{O}(k)$ and $\phi:[x\_0:\ldots:x\_d]\mapsto[x\_0^\ell:\ldots:x\_d^\ell]$ ?



Pushforward of line bundle under "toric isogeny"
It has been proved by Thomsen [Thomsen J. F.. “Frobenius direct images of line bundles on toric varieties” Journal of Algebra 226, no. 2 (2000)] that such a push-forward is a direct sum of line bundles. To be precise, his theorem applies to the Frobenius map on a smooth toric variety in characteristic $p$, but the same result holds for the "toric Frobenius" map you describe. For an explicit description of $\phi\_\*\mathscr{L}$, see Section 2 of [Achinger P. "A Characterization of Toric Varieties in Characteristic p" Int Math Res Notices (2015)]. See also Bondal's article, p. 284 in Oberwolfach Reports from 2006: <https://www.mfo.de/document/0605/OWR_2006_05.pdf> .


In case $X=\mathbb{P}^d$, the description of $\phi\_\* (\mathscr{O}(n))$ is easy to obtain without consulting the above sources. First, note that $\phi^\* \mathscr{L} = \mathscr{L}^q$ for any line bundle $\mathscr{L}$. Second, recall the Horrocks' splitting criterion: a coherent sheaf $\mathscr{F}$ on $\mathbb{P}^d$ is a direct sum of line bundles if and only if $H^i(\mathbb{P}^d, \mathscr{F}(k)) = 0$ for all $0<i<d$ and all $k\in\mathbb{Z}$. Apply this to $\mathscr{F}= \phi\_\* (\mathscr{O}(n))$ and use the projection formula $H^i(\mathbb{P}^d, \phi\_\* (\mathscr{O}(n))(k)) = H^i(\mathbb{P}^d, \phi\_\*(\mathscr{O}(n+qk)) = H^i(\mathbb{P}^d, \mathscr{O}(n+qk)) = 0$ for $0<i<d$ and all $k$. So we know that $\phi\_\* (\mathscr{O}(n)) = \bigoplus\_k \mathscr{O}(k)^{a\_{n,k}}$ for some integers $a\_{n, k}$ which we will now compute. Apply $h^0(-) = \dim H^0(\mathbb{P}^d, -)$ to the above isomorphism to obtain:
$$ \binom{n+d}{d} = \sum\_{k \in \mathbb{Z}} a\_{n,k} \binom{k+d}{d}. $$
Now note $a\_{n, k} = a\_{n-qk, 0}$ (again by the projection formula). Let $a\_m = a\_{m, 0}$, then the above equality says $\binom{n+d}{d} = \sum\_{k \in \mathbb{Z}} a\_{n-qk} \binom{k+d}{d}$. Now if $S(x) = \sum\_{m \in \mathbb{Z}} \binom{m+d}{d}x^m = (1-x)^{-d-1}$, $M(x) = \sum\_{m \in \mathbb{Z}} a\_m x^m$, then the previous equality yields $S(x) = S(x^q)M(x)$, so $M(x) = (1+x+\ldots+x^{q-1})^{d+1}$, thus $a\_{n,k}$ is the coefficient of $x^{n-qk}$ in the polynomial $(1+x+\ldots+x^{q-1})^{d+1}$.


The case of a general toric variety is similar: instead of using the Horrocks criterion, we could have noticed that $\phi$ is the quotient morphism by the action of the $q$-torsion $T[q]$ of the torus $T$, which is a diagonalizable group, and that every line bundle on a toric variety has a $T$-equivariant structure, hence also a $T[q]$-equivariant structure. Thus $\phi\_\* \mathcal{L}$ is the direct sum of its $T[q]$-eigensheaves, which are easily seen to be line bundles.


For example, for $d=1$ and $n=0$ one gets $\phi\_\* \mathscr{O} = \mathscr{O}\oplus\mathscr{O}(-1)^{q-1}$.


