TITLE: What can be said about a projective morphisms that admit decomposition theorem like smooth morphisms?
QUESTION [12 upvotes]: Let $f\colon X\to Y$ be a surjective morphism of smooth projective varieties. If the decomposition theorem for $f$ is given by $$Rf_*\mathbb{C} \simeq \bigoplus_i R^if_*\mathbb{C}[-i],$$ what are the necessary conditions the morphism $f$ must satisfy? Is there an example where such a morphism is not smooth but the decomposition theorem nonetheless looks like the above?
Edit: If I additionally assume that $R^if_*\mathbb{C}$ are local systems for all $i$, can one conclude that $f$ is smooth?  I understand that the limit mixed Hodge structure is pure as there is no monodromy around the singular fibers.

REPLY [2 votes]: Here is an example where $f$ is not smooth but $Rf_* \mathbb{C}$ behaves as if it were:
Let $X$ be a hyperelliptic surface and $f$ the natural morphism to $Y \cong\mathbb{P}^1$. All reduced fibres of $f$ are elliptic curves, but there is a nonzero number of nonreduced fibres, the number dependending on $X$.
The singular cohomology of $X$ is given by $H^0(X, \mathbb{C}) \cong H^4(X,\mathbb{C}) \cong \mathbb{C}$ and $H^1(X, \mathbb{C}) \cong H^3(X, \mathbb{C}) \cong \mathbb{C}^2$. Furthermore, the restriction map $H^1(X, \mathbb{C}) \to H^1(F, \mathbb{C})$ is an isomorphism for any fibre $F$ of $f$.
It is clear that $R^0 f_* \mathbb{C}_X \cong R^2f_* \mathbb{C}_X \cong \mathbb{C}_Y$, so let us consider $R^1f_* \mathbb{C}_X$. Since $H^1(X, \mathbb{C}) \cong \mathbb{C}^2$, we get a natural map $\mathbb{C}^2_Y \to R^1f_* \mathbb{C}$. By evaluating this on stalks and using the proper base change theorem, we see that this is an isomorphism.
Finally, since we know exactly what each sheaf $R^i f_* \mathbb{C}_X$ is, the same proof as in the case $f$ smooth can be used to show that $Rf_*\mathbb{C}_X$ decomposes as a direct sum of its (shifted) cohomology sheaves.
One may ask if a similar statement holds whenever all reduced fibres are smooth (and say $f$ is flat); I did not think about this. It would also be interesting to know if there are examples with non-smooth reduced fibres. Also, note that in the example $R^1 f_* \mathbb{Z}_X$ is not a local system.