Let's say we have a map $f : X \to B$ of complex manifolds, with $B$ simply connected, such that $f$ is diffeomorphic (as a map of smooth manifolds) to the projection $F \times B \to B$. For $b \in B$, we denote its fiber by $X_b:=f^{-1}(b)$. I've seen the following statement:
The singular cohomology spaces $H^i_{sing}(X_b, \mathbb C)$ for $b \in B$ are canonically identified and the fancy way to say this is to say that $R^if_* \underline{\mathbb C}$ is a local system of $\mathbb C$-vector spaces on $B$.
$\underline{\mathbb C}$ of course denotes the constant sheaf with values in $\mathbb C$. I haven't been able to prove this statement, and I couldn't find a reference to look it up. So I was hoping someone could explain:
1) How do we get a canonical identification of the cohomologies of the fibers?
2) How is 1) equivalent to $R^if_* \underline{\mathbb C}$ being a
local system of $\mathbb C$-vector spaces on $B$?
2) An idea is to consider the cartesian diagram $\require{AMScd}$ \begin{CD} X_{b} @>>> F\times B \\ @VVV @VVV \\ \{b\} @>>> B \end{CD}
and then use smooth base change, and the fact that pushforward to the point computes cohomology, to see that the stalk at $b$ of your derived pushforward will be the cohomology of the fiber.