Cohomology of finite covering maps for complex manifold

Question

Let $f:N\to M$ be a finite covering map between connected compact (projective) $n$-dimensional complex manifolds with finite covering group $G$. Let $h ^i:H^i(M,\mathbb{Q})\to H^i(N,\mathbb{Q})$ be the morphism between cohomology groups induced by $f$. Then is there any results about the injectivity and surjectivity of these $h^i$?

Answer 1

In general, $h^i$ is injective, since the transfer homomorphism $\tau \colon H^i(N; \mathbb{Q}) \to H^i(M; \mathbb{Q})$ satisfies $\tau \circ h^i = (\deg f) \mathrm{id}$. In terms of a cocycle $\alpha$ and a cycle $\sigma$ on the appropriate spaces, $$\tau(\alpha)(\sigma) = \sum \alpha(\text{lift of $\sigma$}) .$$ More generally this holds whenever the degree of the finite cover $f$ is a unit in the coefficient ring (here, $\mathbb{Q}$).