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?



When does a morphism between varieties induce surjection/injection morphism on cohomology?
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).


