Let $V$ be a smooth complex projective variety. Choose a very ample class $H\in H^2(V, \mathbb{Q})$. Can there exist finite étale morphisms $\phi\_k:V\to V$ for each $k\geq 1$ such that $\phi^\*\_kH=kH$?



Étale covers pulling back a very ample class to any integer multiple
Yes, this is possible.


Let $E$ be the elliptic curve with equation $y^2 =x^3 -x$, and let $V = E^2$. Then endomorphisms of the abelian surface $V$ are given by two-by-two matrices over $\mathbb Z[i]$.


Let $H$ be the sum of the pullbacks of the class of $O(1)$ (in other words, three times the identity) from both elliptic curves $E$.


Then for every $k$, write $k$ as a sum of four squares $a^2+b^2+c^2+d^2$, and choose the endomorphism $\phi\_k$ given by the matrix


$$ \begin{pmatrix} a+ bi & c+di \\ -c+di & a -bi \end{pmatrix}$$The pullback of the ample class $H$ can be calculated by multiplying the (scalar) matrix of $H$ on the left by this matrix and on the right by its conjugate transpose, which because this matrix times its conjugate transpose is $k$ times the identity, has the effect of multiplying the class of $H$ by $k$.


