111 problems

Let $S$ be a smooth projective surface over the field of complex numbers ${\mathbb{C}}$. Consider its Hilbert scheme of points $\mathsf{Hilb}_n(S)$ and orbifold symmetric product…

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…

Let $X$ be a singular variety admitting crepant resolutions, and let $D^b(\tilde X)$ denote the bounded derived category of coherent sheaves on a crepant resolution $\tilde X$. Cre…

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…

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 $G$ be a finite subgroup of $\operatorname{SL}(n,\mathbb{C})$, and let $\tau:Y\to\mathbb{C}^n/G$ be a crepant resolution. Here $D(Y)$ denotes the derived category of coherent s…

Let $\mathcal{X}$ be a hard-Lefschetz Gorenstein orbifold and let $Y$ be its crepant resolution. Let the GW potentials encode the Gromov–Witten theories of $\mathcal{X}$ and $Y$. B…

Let $X$ be a normal algebraic variety with two crepant resolutions … Here $\operatorname{D}^{\operatorname{b}}(\operatorname{coh}Y_i)$ denotes the bounded derived category of coher…

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…

Physicists' Euler number conjecture. One has

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 $G$ be a finite subgroup of $SL_3(\mathbb{C})$, let $\theta$ be a generic stability condition, and let $\mathcal{M}_{\theta}$ denote the moduli space of $\theta$-stable $G$-con…

Donaldson–Thomas crepant resolution conjecture. One has

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…

Let $X$ be a smooth quasi-projective surface, let $G$ act on $X$, and suppose there is a smooth resolution $\widetilde{X/G}$ of the orbifold $X/G$ such that…

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…

Let $G$ be a finite subgroup of $\operatorname{SL}_3(\mathbb C)$, and let the assumption referred to as condition … holds for every finite subgroup $G$ of…

Let $G$ be a finite subgroup of $\operatorname{SL}_3(\mathbb C)$, let $\pi\colon X\to\mathbb C^3/G$ be the crepant resolution considered in the paper, and let the complex construct…

Let $G$ be a finite subgroup of $\operatorname{SL}_3(\mathbb C)$, let $\pi\colon X\to\mathbb C^3/G$ be a crepant resolution, and let $\{\mathcal R_k\}$ and $\{S_k\}$ be the tautolo…

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

Let $\mathcal{X}$ be the orbifold and $Y$ its crepant resolution, with big quantum products parametrized respectively by $\tau\in H(\mathcal{X})$ and $t\in H(Y)$. Together with the…