65 problems

Let $X$ be a quasi-projective variety with an action of a finite group $G$, let $X/G$ have Gorenstein singularities, and let $Y$ be any crepant resolution of $X/G$. Denote by…

Nakamura's conjecture. Then $Y$ is a crepant resolution of $\mathbb{C}^r/G$. This conjecture extends the three-dimensional result of Bridgeland, King, and Reid to higher dimensions…

Let $S$ be a rational homology projective plane with quotient singularities, and write … The smooth locus $S^0$ is assumed to be simply connected. Kollár's algebraic Montgomery–Yan…

Let $X$ and $Y$ be projective varieties with only finite quotient singularities, and let $\mathcal{X}(X)$ and $\mathcal{X}(Y)$ be their associated stacks. The varieties $X$ and $Y$…

Shokurov's index conjecture. There exists a positive integer $r(n,a)$ such that, whenever $\operatorname{mld}_p(X)=a$, the Cartier index of $K_X$ at $p$ is at most $r(n,a)$.

Let a finite quotient singularity be a singularity obtained as a quotient by a finite group action. Riemenschneider's conjecture. Finite quotient singularities should deform to fin…

Let $S$ be a rational homology projective plane, meaning a normal projective complex surface whose Betti numbers are the same as those of $\mathbb{CP}^2$. Assume that $S$ has quoti…

Let $V$ be a vector space and let $G\subset SL(V)$ be finite. Set $X=V/G$, and assume that $X$ admits a smooth crepant resolution $Y\to X$. For a cyclic subgroup…

Let $X=V/G$ be the quotient of a complex vector space $V$ by a finite subgroup $G\subset SL(V)$, and assume that $X$ admits a smooth crepant resolution $Y\to X$. Write…

GPSS conjecture. All such cyclic Gorenstein quotient singularities admit torus-equivariant projective, crepant, full resolutions. The theorem immediately preceding the conjecture p…

Let $S$ be a rational homology $\mathbb C\mathbb P^2$ with quotient singularities, and write … Assume that $S^0$ is simply connected. Simply connected smooth-locus conjecture. Then…

Let $G\subset\operatorname{SL}(r,\mathbb{C})$ be finite, and consider quotient complete-intersection singularities $\mathbb{C}^{r}/G$ in arbitrary dimension. Existence conjecture.…

Let $G$ be a finite subgroup of $\operatorname{SL}(r,\mathbb{C})$, let $X=\mathbb{C}^{r}/G$, and let $\widehat{X}\longrightarrow X$ be a crepant desingularization. For each integer…

Let $\Gamma<\operatorname{SL}(2)$ be a finite subgroup, let $\Gamma_n=\Gamma\wr\mathcal S_n$, and set … A projective or proper symplectic resolution is a projective, respectively p…

Higher-dimensional Terminal Lemma. For all sufficiently large $d\in\mathbb{N}$, if the Shokurov minimal log-discrepancy is at least

Vasyunin's conjecture. Up to a permutation of the coordinates in $T^{(2d+1)}$, the corresponding points lie in the subtorus

Vasyunin's completeness conjecture. Vasyunin's lists for $d=3$ and $d=4$ are complete.

Let $S$ be a projective surface with quotient singularities such that … and let $S^0$ denote its smooth locus. Assume that $S^0$ is simply connected. Rationality conjecture. Then…

Let $G$ be a finite group acting on $\a9^4$ freely away from the origin, and consider the quotient singularity $\a9^4/G$. A nicest partial resolution is a partial resolution of thi…

Let $G \subseteq \operatorname{SL}_3(\mathbb{C})$ be a finite subgroup, let $X=\mathbb{C}^3/G$, and let $\Pi$ be the space of stability parameters for $G$-constellations. A $G$-con…

Let $X$ be a compact complex manifold with an action of a finite group $G$, and assume that for every $1\ne g\in G$ the fixed locus $X^g$ has codimension at least $2$. Let $[X/G]$…

Let $G$ be a finite subgroup of $\operatorname{SL}(d,\mathbb{C})$ such that $\mathbb{C}^d/G$ is minimally embeddable as a complete intersection in an affine space $\mathbb{C}^r$, w…

Smooth G-Hilbert-scheme conjecture. The quotient construction should be nonsingular, or crepant over $\mathbb C^n/G$, for groups generated by subgroups contained in…

Tautological-sheaf basis conjecture. Under appropriate circumstances, the tautological sheaves $\mathcal F_\rho$ form a $\mathbb Z$-basis of $K_0(\operatorname{Coh}Y)$; their Chern…

McKay correspondence conjecture. There should be natural bijections