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?