Local mirror symmetry for general Calabi–Yau threefolds

Let $Y=(Y,J,h)$ be a general Calabi–Yau threefold and let $\chi\in\mathcal{K}_{\mathbb{C}}(Y)$ satisfy $\operatorname{Im}\chi=-\frac12\operatorname{Im}h$. Call $(Y,\chi)$ a polarized couple. A mirror polarized couple is $(Y^{\circ},\chi^{\circ})$, with $Y^{\circ}=(Y^{\circ},J^{\circ},h^{\circ})$ sufficiently general and $\operatorname{Im}\chi^{\circ}=-\frac12\operatorname{Im}h^{\circ}$. Local mirror symmetry conjecture. There exist such a mirror polarized couple, open neighborhoods $\mathcal{U}\subset\mathcal{M}_Y$ and $\mathcal{U}^{\circ}\subset\mathcal{M}_{Y^{\circ}}$ with local product structures $\mathcal{U}\cong U_{\mathbb{C}}\times U_K$ and $\mathcal{U}^{\circ}\cong U^{\circ}_{\mathbb{C}}\times U_K^{\circ}$, and a biholomorphism $m:\mathcal{U}\to\mathcal{U}^{\circ}$ reversing the factors, so that $m(U_{\mathbb{C}})=U_K^{\circ}$, $m(U_K)=U_{\mathbb{C}}^{\circ}$, and $d_{(J,\chi)}m=\mu'\times\mu”$. This gives a local mirror map exchanging complex and complexified Kähler moduli; the source invokes generic local triviality in dimension three but does not establish the conjecture in general.