Let $f:\mathcal{C} \to \Delta^*$ be a family of smooth, projective curves over a punctured disc. Denote by $\mathbb{H}^1:=R^1f_*\mathbb{Z}$ the associated local system, such that for every $t \in \Delta^*$, the fiber $\mathbb{H}^1_t \cong H^1(\mathcal{C}_t,\mathbb{Z})$. My question is: Is there a Poincare duality associated to the local system? In particular, does there exist an isomorphism of sheaves: $$\mathbb{H}^1 \to \mathcal{H}om(\mathbb{H}^1, \mathbb{Z}_{\Delta^*}),$$ where $\mathcal{H}om$ denotes sheaf homomorphism and $\mathbb{Z}_{\Delta^*}$ is the constant sheaf on $\Delta^*$ with stalk $\mathbb{Z}$. Any reference will be most welcome.


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