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 $\phi^*_kH=kH$?
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$\begingroup$ Complementing Will's excellent answer - I'm unable to find a reference right now, but I imagine that the following is true and known: if $V$ is a smooth projective algebraic variety admitting an etale endomorphism $f\colon V\to V$ which is polarized in the sense that $f^* H = kH$ for some ample class $H$ and some integer $k$, then $V$ is an abelian variety. $\endgroup$– Piotr AchingerCommented Jul 27, 2020 at 16:26
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$\begingroup$ On second thought, the assertion perhaps should be that $V$ admits a finite etale (Galois) cover $A\to V$ with $A$ an abelian variety. And of course $k>1$. $\endgroup$– Piotr AchingerCommented Jul 27, 2020 at 16:34
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$\begingroup$ Here's a attempt to prove Piotr's statement. As $V$ has no rational curve (otherwise $f$ would be isomorphic over it, but $H$ has necessarily degree $0$ over it). So $K_V$ is nef, but $K_V\cdot H^{n-1}=0$ since $f^*K_V=K_V$. Hence $c_1(V)=0$. By Beauville-Bogomolov, $V$ admits an étale cover $A\times W\to V$, with $A$ is abelian and $\pi_1(W)=0$. Let $g$ be the étale cover obtained on $A\times W$ by fiber product. One checks that the fundamental group acts diagonally on $\mathbb C^{\dim A}\times W$ hence $g=g_A\times g_W$ is product. Since ${\rm Pic}(A\times W)$ is a product, one has $W=0$. $\endgroup$– HenriCommented Jul 27, 2020 at 21:40
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$\begingroup$ Alternative proof once one has $c_1(V)=0$: the same argument shows $c_2(V)\cdot H^{n-2}=0$ hence by Yau's uniformization theorem, $V$ is a torus quotient. $\endgroup$– HenriCommented Jul 28, 2020 at 7:31
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$\begingroup$ @Henri do you need equality in the rational singular cohomology or in the Picard group for the argument to work? $\endgroup$– user145520Commented Jul 28, 2020 at 7:43
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1 Answer
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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 $k$.
answered Jul 27, 2020 at 15:27