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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?

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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.)

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  • $\begingroup$ Thank you. What if consider the localy system $R^2\pi_*(\mathbb{Z})$ modulo torsion, i.e. its fiber at $t$ is $H^2(X_t,\mathbb{Z})$ (modulo torsion)? Does the monodromy of this example come from the torsions of $H^2(X_t,\mathbb{Z})$ ? $\endgroup$
    – Wenzhe
    Oct 21 '17 at 17:35
  • $\begingroup$ No, $H^2$ of a cubic surface is torsion free. $\endgroup$
    – abx
    Oct 22 '17 at 5:59

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