Hyper-Kähler resolution conjectures for orbifold invariants

Let $\mathcal{X}$ be a smooth algebraic orbifold with coarse moduli space $|\mathcal{X}|$, and suppose there is a crepant resolution $Y\to |\mathcal{X}|$ such that $Y$ is hyper-Kähler. Here $H^*_{\operatorname{orb}}(\mathcal{X},\mathbb{C})$, $K^{\operatorname{orb}}_0(\mathcal{X})$, $\operatorname{CH}^*_{\operatorname{orb}}(\mathcal{X})$, and $M_{\operatorname{orb}}(\mathcal{X})$ denote the corresponding orbifold cohomology, K-theory, Chow ring, and motive; $K_0(Y)_{\mathbb{C}}$ and $\operatorname{CH}^*(Y)_{\mathbb{C}}$ are the complexifications, and $M(Y)$ is the mixed motive of $Y$. Hyper-Kähler resolution conjectures. There are isomorphisms

$$H^*(Y,\mathbb{C})\simeq H^*_{\operatorname{orb}}(\mathcal{X},\mathbb{C}),$$

of graded commutative $\mathbb{C}$-algebras;

$$K_0(Y)_{\mathbb{C}}\simeq {K^{\operatorname{orb}}_0}(\mathcal{X})_{\mathbb{C}}^{\wedge},$$

of commutative $\mathbb{C}$-algebras;

$$\operatorname{CH}^*(Y)_{\mathbb{C}}\simeq \operatorname{CH}_{\operatorname{orb}}^*(\mathcal{X})_{\mathbb{C}},$$

of commutative graded $\mathbb{C}$-algebras; and

$$M(Y)\simeq M_{\operatorname{orb}}(\mathcal{X}),$$

as commutative algebra objects in the category of complex mixed motives $\operatorname{DM}_{\mathbb{C}}$. These conjectures seek to identify orbifold theories with the corresponding theories of hyper-Kähler crepant resolutions; the source presents them as a series of conjectural extensions of Ruan's cohomological proposal, with higher K-theoretic and Chow-theoretic versions subsequently proposed.