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.



What can be said about a projective morphisms that admit decomposition theorem like smooth morphisms?
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.


