Let $f: X\to Y$ be a submersion between projective manifolds. Then $Rf^i\_\*(\mathbb{C}\_X)$ are local systems on $X$, for all $i$. 


My question is whether the converse is true. More precisely, let $f: X\to Y$ be a holomorphic map between projective manifolds. Suppose that for all $i$, $Rf^i\_\*(\mathbb{C}\_X)$ are global local systems on $X$. Is $f$ necessarily a submersion?


If this is not true, what is a counter-example?



When is a proper map between smooth varieties a submersion?
I am just posting my comment above as an answer. There are counterexamples that depend on the fact that the divisible coefficient sheaf $\mathbb{C}\_X$ (for the analytic topology, presumably) does not detect certain torsion phenomena. Begin with a smooth, projective, geometrically integral curve $E$ of genus $1$ and a nontrivial $2$-torsion translation $\tau:E\to E$. Form $Z=E\times \mathbb{A}^1$. Let a cyclic group $G$ of order $2$ act diagonally on $Z$ via $\tau$ on $E$ and via $t\mapsto -t$ on $\mathbb{A}^1$. Let $X$ be $Z/G$, and let $Y$ be $\mathbb{A}^1/G$. Let $f:X\to Y$ be the unique morphism induced by the projection $\text{pr}\_2:E\times \mathbb{A}^1 \to \mathbb{A}^1$. Then $f\_\*\mathbb{C}\_X$ is $\mathbb{C}\_Y$, $R^2f\_\*\mathbb{C}\_X$ is isomorphic to $\mathbb{C}\_Y$, and $R^1f\_\*\mathbb{C}\_X$ is also a localy system isomorphic to $\mathbb{C}\_Y^{\oplus 2}$. It is the last one that is a bit more delicate, since for every $y \in Y$ other than the origin, for the fiber $X\_y=f^{-1}(\{y\})$, the map on homology, $$H\_1(X\_y;\mathbb{Z})\to H\_1(X;\mathbb{Z}),$$ is *not* an isomorphism. It is injective and the image is an index $2$ subgroup. However, once we tensor with the divisible group $\mathbb{C}$, the induced map is an isomorphism. To say it differently, $R^1f\_\*\mu\_2$ is not a local system.


