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

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 $$f$$?

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).
• More generally, it happen quite often that $\mathcal{O}_Y\to Rf_* \mathcal{O}_X$ splits in the derived category, and then so does $L\to Rf_* f^* L$, so the pull-back maps $H^p(Y, L)\to H^p(X, f^* L)$ are injective. – Piotr Achinger Apr 13 at 9:06