1
$\begingroup$

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 $f$?

$\endgroup$
1
$\begingroup$

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).

| cite | improve this answer | |
$\endgroup$
  • 1
    $\begingroup$ 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. $\endgroup$ – Piotr Achinger Apr 13 at 9:06
  • $\begingroup$ @Piotr Achinger, what do you mean by "quite often"? $\endgroup$ – manifold Apr 13 at 22:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.