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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 $\phi:[x_0:\ldots:x_d]\mapsto[x_0^\ell:\ldots:x_d^\ell]$ ?

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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: $$ \binom{n+d}{d} = \sum_k a_{n,k} \binom{k+d}{d}. $$ 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 $\binom{n+d}{d} = \sum_k a_{n-qk} \binom{k+d}{d}$. Now if $S(x) = \sum_m \binom{m+d}{d}x^m = (1-x)^{-d}$, $M(x) = \sum_m a_m x^m$, then the previous equality yields $S(x) = S(x^q)M(x)$, so $M(x) = (1+x+\ldots+x^{q-1})^d$, thus $a_{n,k}$ is the coefficient of $x^{n-qk}$ in the polynomial $(1+x+\ldots+x^{q-1})^d$.

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 $\phi_* \mathscr{O} = \mathscr{O}\oplus\mathscr{O}(-1)^{q-1}$.

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