Let $f:\mathcal{X} \to \Delta$ be a flat family of projective varieties, smooth over the punctured disc $\Delta^*$ and the central fiber is a simple normal crossings divisor. Let $\mathcal{Z} \subset \mathcal{X}$ be a closed subscheme of relative codimension, say $p$, and flat over $\Delta$. Does the central fiber $\mathcal{Z}_0$ of $\mathcal{Z}$ define an element, say $\eta$ in $H^{2p}(\mathcal{X}_0,\mathbb{C})$ such that under the specialization morphism $H^{2p}(\mathcal{X}_0,\mathbb{C}) \to H^{2p}(\mathcal{X}_t,\mathbb{C})$ for $t \in \Delta^*$ general, $\eta$ maps to the cohomology class of $[\mathcal{Z}_t] \in H^{2p}(\mathcal{X}_t,\mathbb{C})$? Any reference will be most welcome.


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