Suppose $\pi:X \rightarrow \Delta$ is a fibration and $\pi^{-1}(0)$ is the only singular fiber, and let's also assume that all the fibers are complex projective varieties. Then we have a fibration between smooth manifolds,
\begin{equation}
\pi: X \setminus X\_0 \rightarrow \Delta^\*
\end{equation}
For every smooth fiber $X\_t$, let's assume that $h^{2,0}(X\_t)=h^{0,2}(X\_t)=0$, then the pure Hodge structure on $H^2(X\_t,\mathbb{Z})$ is isomorphic to $\mathbb{Z}(-1)^{k}$, where $k=h^{11}(X\_t)$. Naively it seems that the Hodge structures on $H^2(X\_t,\mathbb{Z})$ does not vary when you vary the parameter $t$, so my question is about this.


The local system $R^2\pi\_\*(\mathbb{Z})$ on $\Delta^\*$ has fiber over $t\in \Delta^\*$ as $H^2(X\_t,\mathbb{Z})$. Is the monodromy of $R^2\pi\_\*(\mathbb{Z})$ trivial in general?



Monodromy of a one parameter fibration
No, $\pi:X\to\Delta$ could be a family of cubic surfaces and then the monodromy will be generated by an element $w$ of the Weyl group $W(E\_6)$.
If the total space $X$ is smooth and the closed fiber $X\_0$ has a singularity of type $E\_6$ then $w$ is a Coxeter element. (This is a result of Demazure but I don't have the reference to hand.)


