The refined crepant transformation conjecture

Let $\mathscr{L}^{\mathcal{X}}\subset\mathscr{V}^{\mathcal{X}}$ and $\mathscr{L}^{\mathcal{Y}}\subset\mathscr{V}^{\mathcal{Y}}$ be the equivariant genus-zero Gromov–Witten Lagrangian cones, and let $\mathbb{U}:\mathscr{V}^{\mathcal{X}}\to\mathscr{V}^{\mathcal{Y}}$ be the symplectic transformation in the crepant transformation conjecture. Let $\mathscr{V}^{\mathcal{X}}_c$ and $\mathscr{V}^{\mathcal{Y}}_c$ denote the subspaces spanned by compactly supported classes, and let $\mathbb{1}_g$ and $\tilde{\mathbb{1}}_g$ be the fundamental classes of the corresponding inertia components. The refined crepant transformation conjecture. The crepant transformation conjecture holds. In addition, $\mathbb{U}$ has coefficients in $\mathbb{C}[\lambda,z,z^{-1}]$; in the non-equivariant limit it restricts to an isomorphism between $\mathscr{V}^{\mathcal{X}}_c$ and $\mathscr{V}^{\mathcal{Y}}_c$; and

$$\mathbb{U}(\mathbb{1}_g)=C_0(\lambda)\tilde{\mathbb{1}}_g+\sum_{b=1}^{d-1}(\lambda+H)\cdot C_b(\lambda)\tilde{\mathbb{1}}_{g\mathfrak{j}^b},$$

where $C_b(\lambda)\in H^*(\mathcal{Y})[\lambda]((z^{-1}))$. In particular, the restriction $\mathbb{U}_c$ of $\mathbb{U}$ to $\mathscr{V}^{\mathcal{X}}_c$ has image in the $\mathbb{C}((z^{-1}))$-span of $(\lambda+H)\cdot H^*_{CR}(\mathcal{Y})[\lambda]$. These coefficient, compact-support, and image conditions refine the expected Fourier–Mukai-type identification; the source describes them as natural and gives no resolution of the conjecture.