If $p : X \rightarrow Y$ is the dominant surjective finite morphism of varieties between smooth projective varieties over of same dimension over $\mathbb{C}$, then do we know about the properties of the pull-back maps $p^{\star} : H^{k,k}(Y) \rightarrow H^{k,k}(X)$ is surjective/injective? If someone could provide reference to this?
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For a finite map between smooth varieties, the induced pullback map on cohomology is injective (hence it is somewhat rarely surjective - only when it's an isomorphism).
This is because there a is a trace map $H^{k,k}(X) \to H^{k,k}(Y)$, and the composition of the two $H^{k,k}(Y) \to H^{k,k}(X) \to H^{k,k}(Y)$ is multiplication by the degree of $p$.
This holds in the greater generality of any surjective morphism between smooth projective varieties, but there it requires more powerful tools to prove (it follows from the decomposition theorem, but I'm not sure if that's really needed).
answered Feb 17, 2022 at 21:14
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3$\begingroup$ The pushforward map is always surjective (and rarely injective); the pullback map is always injective (and rarely surjective). $\endgroup$ Commented Feb 17, 2022 at 21:34
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$\begingroup$ @wnx I somehow managed to screw up the words... $\endgroup$ Commented Feb 17, 2022 at 21:42
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4$\begingroup$ @WillSawin For the "greater generality" part, no need for the decomposition theorem: just cut by hyperplanes to get a smooth $Z\subset X$ so that $Z\to Y$ is generically finite, then your previous argument works. $\endgroup$ Commented Feb 17, 2022 at 22:35
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$\begingroup$ @wnx Torsion classes may be sent to zero, as happens for the double cover of an Enriques surface by a K3 surface. $\endgroup$ Commented Feb 18, 2022 at 17:31