Equivariant asymptotic Hodge conjecture for Calabi–Yau threefolds

Let $\pi:\mathcal{Y}\to\Delta$ be a $G$-equivariant semistable degeneration of Calabi–Yau threefolds with ordinary double points, and let $\widetilde{Y}_0$ be a $G$-equivariant crepant resolution of $Y_0$. The group $G$ acts fibrewise, and $\beta\in H^{2,2}(Y_t,\mathbb{Q})^G$ denotes a $G$-invariant rational $(2,2)$-class; $\mathrm{sp}$ is the specialization map and $\mathrm{CH}^2(\widetilde{Y}_0)_{\mathbb{Q}}^G$ denotes the group of rational algebraic codimension-two $G$-cycles. Equivariant asymptotic Hodge conjecture. Every such class is the limit of an algebraic $G$-cycle of codimension $2$ on $\widetilde{Y}_0$; equivalently,

$$\beta\in H^{2,2}(Y_t,\mathbb{Q})^G \quad\Longrightarrow\quad \mathrm{sp}(\beta)\in \mathrm{CH}^2(\widetilde{Y}_0)_{\mathbb{Q}}^G,$$

and $\beta$ can be transported back to $Y_t$ as an algebraic $G$-cycle via the Clemens–Schmid isomorphism. This extends the proposed node-degeneration method from K3 surfaces to equivariant Calabi–Yau threefolds; the supplied text presents it as a conjectural analogue, without evidence of resolution.