Let $X$ be a smooth projective rigid Calabi-Yau threefold.
Question. Does there exist a finite map $X\to X$ of degree $>1$?
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$\begingroup$ The article "Endomorphisms of smooth projective 3-folds with non-negative Kodaira dimension" Publ. Res. Inst. Math. Sci. 38 (2002), no. 1, 33–92 by Y. Fujimoto probably contains the answer to your question. $\endgroup$– nafCommented Apr 30, 2021 at 6:02
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3$\begingroup$ Welcome @smprcy. Is a CY threefold simply connected with trivial canonical bundle with your definitions? If so, you can argue as follows: your finite map will be etale (by Riemann-Hurwitz) and thus trivial by simple-connectedness. $\endgroup$– Ariyan JavanpeykarCommented Apr 30, 2021 at 6:06
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$\begingroup$ @AriyanJavanpeykar Could you explain the argument in more detail? Don't some K3s have non-trivial self-isogenies? $\endgroup$– nrlofCommented Apr 30, 2021 at 6:53
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3$\begingroup$ @nrlof: no. They have "self-rational maps", but not everywhere defined. Indeed any finite map between smooth projective varieties with trivial canonical bundle must be étale. $\endgroup$– abxCommented Apr 30, 2021 at 13:00
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1 Answer
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One can show that such a map in the question doesn't exist (no need to assume simply-connectedness). As abx pointed out, any finite map between smooth projective varieties with trivial canonical bundle must be étale.
Let $\chi(X)$ be the topological Euler characteristic of $X$ and $d$ be the degree of the map. Since $X$ is rigid, $$\chi(X) = 2 \left( h^{1,1}(X)- h^{1,2}(X) \right) =2 h^{1,1}(X) > 0.$$
Noting $d \cdot \chi(X) = \chi(X)$, one has $d=1$.
