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 $f$?