Injectivity of the cohomology map associated to the pullback of line bundles

Question

Let $f:X\to Y$ be a flat, surjective, smooth morphism between smooth algebraic varieties (over $\mathbb C$). We assume that $f$ has relative dimension $n$ and we assume also that $\dim Y\ge 2$ (just to avoid the case of a curve that might be easier).

Let $L$ be a line bundle on $Y$, then we have a homomorphism in sheaf cohomology:

$$H^p(Y,L)\to H^p(X, f^\ast L) \quad\text{for } p=0,1,\ldots,\dim Y $$

Can we say anything about the injectivity of this map? Do we need some additional condition on $f$?

Answer 1

The unit of adjunction map $L\rightarrow f_\ast f^\ast L$ is an isomorphism if the fibres of $f$ are connected (I assume this in what follows). Then the Leray spectral sequence $E^{pq}_2=H^p(R^q f_\ast f^\ast L)\Rightarrow E^{p+q}=H^{p+q}(f^\ast L)$ has edge maps $e^p:E^{p0}\rightarrow E^p$ which are exactly the maps you are interested in. In particular, these maps are isomorphisms if $f$ is finite. In general $e^0$ is an isomorphism, $e^1$ is injective, but $e^2$ need not be injective (exact sequence of low degree terms). If $f$ is of the form $X\times Y\rightarrow Y$ then the maps $e^p$ are certainly injective (Künneth).