Let $M$ be a closed connected complex manifold with $\mathrm{dim}\:M=n$. Can there exist holomorphic covering maps $\phi_k:M\to M$ for all integers $k\geq 1$ such that $\phi_k^*:H^n(M, \mathbb{Z})\to H^n(M, \mathbb{Z})$ is multiplication by $k$?
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2$\begingroup$ You can't mean literal 'covering maps'. Just take $M = \mathbb{P}^1$, which has no nontrivial covering maps of this kind. Maybe you mean 'branched covering maps'? Even in this case, it's not generally possible. $\endgroup$– Robert BryantCommented Aug 17, 2020 at 19:52
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1$\begingroup$ Elliptic curves have many maps like these; take a rectangle in the plane, identify opposite sides, and map each complex number by an integer multiple. $\endgroup$– Ben McKayCommented Aug 17, 2020 at 20:01
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$\begingroup$ @RobertBryant you are correct that there no covering maps for a simply-connected manifold. But I am looking for just one example. $\endgroup$– user145520Commented Aug 17, 2020 at 20:07
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2$\begingroup$ It's not true that the degrees of the self-maps of an elliptic curve are necessarily square integers; it depends on the elliptic curve. For example, for the square torus, the degree of a self-map can be the sum of any two square integers: Multiplication by a Gaussian integer $a+b\,i$ preserves the lattice of Gaussian integers $\mathbb{Z}[i]\subset\mathbb{C}$ and induces a self-map of degree $a^2+b^2$ on the square torus. More generally, for an elliptic curve that admits a complex multiplication, the degrees of its self-maps are the values of a quadratic form on an integer lattice of rank $2$. $\endgroup$– Robert BryantCommented Aug 19, 2020 at 9:19
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3$\begingroup$ If $M = \mathbb{C}^n/\Lambda$ where $\Lambda$ is a lattice, let $R\subset M_n(\mathbb{C})$ be the ring satisfying $R(\Lambda)\subseteq\Lambda$. Then $R$ is additively a free abelian group of rank at most $2n^2$, and $D(r) = \det_\mathbb{R}(r)$ for $r\in R$ is a nonnegative integer-valued form of degree $2n$ on $R$. Given any nonnegative integer $m$, one can choose $n$ and $\Lambda$ so that the multiplicative set $D(R)\subset\mathbb{Z}$ contains $\{1,2,\ldots, m\}$. One can even arrange that $M$ be a product of elliptic curves. Might there be a $\Lambda$ where $D(R)$ contains $\mathbb{Z}^+$? $\endgroup$– Robert BryantCommented Aug 24, 2020 at 16:00
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