Let $(X,T)$ be a smooth complex toric variety of dimension $n$ 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 $\phi:[x_0:\ldots:x_d]\mapsto[x_0^\ell:\ldots:x_d^\ell]$ ?

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