When is a proper map between smooth varieties a submersion? 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?
 A: 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.
