Generalization of the Leray-Hirsch theorem

Question

We know the classical Leray-Hirsch theorem for fibrations. My question is, whether a similar statement also holds for flat, proper morphism? In particular, consider a faithfully flat, proper morphism $f:Y \to X$ with both $X$ and $Y$ non-singular, irreducible varieties over $\mathbb{C}$. Can we say that if for every $x \in X$, $H^q(f^{-1}(x), \mathbb{Q})$ is the same, then $R^qf_*\mathbb{Q}$ is a local system? More generally if for every $q$, $H^q(f^{-1}(x),\mathbb{Q})$ does not depend on the choice of $x$, then can we write $H^q(Y,\mathbb{Q})$ as a direct sum of $H^i(f^{-1}(x),\mathbb{Q}) \otimes H^{q-i}(X,\mathbb{Q})$ as $i$ varies? Any reference will be most welcome.

Answer 1

Here's a version of Leray-Hirsch for a proper morphism not a priori assumed to be a fibration.

Suppose that:

• $f \colon X \to Y$ is a proper morphism of smooth varieties,
• all fibers of $f$ have the same Betti numbers,
• for a generic point $y$ of $Y$, the restriction map $H^\ast(X,\mathbf Q) \to H^\ast(f^{-1}(y),\mathbf Q)$ is surjective ("Leray-Hirsch").

Then all sheaves $R^qf_\ast \mathbf Q$ are trivial local systems, the Leray spectral sequence degenerates, and $H^\ast(X,\mathbf Q) \cong H^\ast(Y,\mathbf Q) \otimes H^\ast(F,\mathbf Q)$ where $F$ denotes any fiber of $f$.

Proof: By the decomposition theorem, $Rf_\ast\mathbf Q$ is a sum of shifted perverse sheaves. Choose a dense open $U \subset Y$ over which these are local systems. This open subset contains the generic point $y$, so by the usual Leray-Hirsch principle these local systems are all trivial over $U$. The intermediate extension of these local systems gives a summand of $Rf_\ast\mathbf Q$, but the intermediate extensions are again trivial. By the assumption on the Betti numbers of the fibers there can be no further summands inside $Rf_\ast\mathbf Q$, and we are done.

Remark: I do not know an example of a morphism $f$ satisfying the assumptions of the argument, but which is not in fact a fibration.

Answer 2

For any reasonable interpretation of the terms "$H^q(f^{-1}(x), \mathbb{Q})$ does not depend on $x$" or "...is the same", $R^qf_*\mathbb{Q}$ would be a local system. So then by Deligne, the Leray spectral sequence degenerates to give isomorphisms $$ H^q(Y,\mathbb{Q}) = \bigoplus_i H^i(Y, R^{q-i}f_*\mathbb{Q})$$ If the monodromies of the all the above local systems are trivial, then you would get the sort of isomorphisms that you are asking for, but not otherwise.