50 problems

Let $M$ be a smooth projective variety endowed with a faithful action of a finite group $G$, and let $X:=M/G$ be its quotient. Assume that $X$ is Gorenstein, and let $Y\to X$ be an…

Let $\mathcal{R}$ be an ADE root system, let $\mathcal{R}^{\mathrm{fold}}$ be its folding, and let $\mathcal{X}_{\mathcal{R}^{\mathrm{fold}}}$ be the corresponding orbifold with cr…

Let $X$ be a variety admitting crepant resolutions, including commutative resolutions and non-commutative crepant resolutions. Van den Bergh's conjecture. All crepant resolutions o…

Let $R$ be a Gorenstein toric algebra, meaning the coordinate ring associated to a Gorenstein cone. An NCCR of $R$ is an algebra of the form $\Lambda=\operatorname{End}_R(M)$ for…

Let $\mathcal{X}$ be an orbifold with coarse moduli space $X$, and let $\phi\colon Z\to X$ be a crepant resolution. Iritani's conjecture. There exists an isomorphism … induced by a…

Let $X$ be a normal algebraic variety with Gorenstein singularities, and let … be two crepant resolutions by schemes or Deligne–Mumford stacks. An equivalence of triangulated categ…

Let $R$ be an equidimensional normal Gorenstein ring. A noncommutative crepant resolution of $R$ is an endomorphism ring of a modifying $R$-module with finite global dimension, whi…

Let $R$ be a Gorenstein $coperatorname{\mathbb{C}}$-algebra. A crepant resolution of $R$ is a crepant resolution of the corresponding singularity, and an NCCR of $R$ is a non-commu…

Let $Y$ be a Calabi–Yau orbifold and let $\widetilde{Y}\to Y$ be a crepant resolution. Orbifold–resolution conjecture. There is a constant $C(\widetilde{Y}/Y)$, depending only on t…

Let $\mathcal{Z}$ be a hard-Lefschetz $3$-orbifold, and let … be a crepant resolution. Let $DT_{mr}(-)$ denote the reduced, multi-regular Donaldson–Thomas potential, and let…

Reid's derived McKay correspondence conjecture. There is an equivalence

Let $V$ be a vector space, let $G\subset SL(V)$ be a finite subgroup, and set $X=V/G$. Assume that there is a proper smooth crepant resolution $\pi:Y\to X$. For each homomorphism…

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 $G$ be any finite subgroup of $\operatorname{SL}(3,\mathbb{C})$, and let $\mathcal{H}^G(\mathbb{C}^3)$ denote the $G$-Hilbert scheme. Nakamura's conjecture. The space…

Let $\mathcal{X}$ be an orbifold satisfying the hard Lefschetz condition and admitting a crepant resolution $Y$. Let $F^Y$ and $F^\mathcal{X}$ denote their potential functions, wit…

Let $Y$ be a singular variety admitting crepant categorical resolutions, and call a categorical resolution indecomposable when it has no nontrivial decomposition in the relevant ca…

Let $A=\operatorname{Spec}\bigoplus_{n\geq0}H^0(w\mathbb{P},-nK)$ be the anticanonical affine cone over the weighted projective space $w\mathbb{P}$, and let $\widetilde{A}\to A$ be…

Let $V$ be a finite-dimensional vector space over a field $k$, and let $E$ be a finite subgroup of $\operatorname{GL}(V)$ whose order is invertible in $k$. A crepant resolution is…

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 $Y$ be a variety with transversal $A_n$-singularities, $n\geq 2$, and let $\rho:Z\to Y$ be its crepant resolution. Let $K$ be the line bundle on the singular locus appearing in…

Let $G\subset \operatorname{Sl}_n(\mathbb C)$ be a finite group, put $W=\mathbb C^n/G$, and let $X\rightarrow W$ be a crepant resolution. Write $D^b_G(\mathbb C^n)$ for the bounded…

Let $X$ be a smooth Calabi–Yau manifold with an action of a finite group $H$ preserving the holomorphic volume form, let $Y=X/H$, and let $h_{st}^{p,q}(Y)$ denote its stringy Hodge…