54 problems

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…

Physicists' Euler number conjecture. One has

Let $G$ be a finite small cyclic subgroup of $\operatorname{GL}(2,\mathbb{C})$, and let $\operatorname{Hilb}^G(\mathbb{C}^2)$ be the $G$-Hilbert scheme. The irreducible representat…

Let $\Gamma$ be a finite subgroup of $SL_2(\mathbb{C})$, let $\Gamma_n=\Gamma^n\rtimes S_n$, and let $X_{\Gamma,n}$ be the union of connected components of the $\Gamma$-fixed locus…

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 $X=V/G$ 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…

Let $X=M/G$, where $M$ is a quasi-projective variety of dimension $n$ and $G$ is finite, and let $\mathcal{X}$ be the associated stack. Let $Y$ be the component of the $G$-Hilbert…

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…

Twisted McKay correspondence. In the situation described above, for every $\alpha\in B_G(U)$ there exists a derived equivalence

Let $B_n$ be the tautological bundle on the Hilbert scheme of points $\operatorname{Hilb}^n(\mathbb{C}^2)$, and let $\psi^jB_n$ denote its $j$-th Adams power. Under the McKay corre…

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 $V$ be a finite-dimensional symplectic vector space, let $G\subset Sp(V)$ be a finite group, and let $X\to V/G$ be a crepant resolution. The orbifold cohomology…

Let $M$ be an algebraic symplectic manifold with a symplectic action of a finite group $G$, and let $X\to M/G$ be a smooth symplectic resolution. Choose deformation-quantizations…

Let $M$ be a holomorphic symplectic manifold with a finite symplectic $G$-action, and let $X\to M/G$ be a symplectic resolution. Define … where the shifted term is the direct sum o…

Let $V$ be a symplectic vector space, let $G$ act symplectically on $V$, and let $X\to V/G$ be a symplectic resolution. The natural dilation action of $\mathbb C^*$ on $V$ lifts to…

K-orbifold string theory conjecture. There is a natural isomorphism

Let $G\subset \operatorname{SL}(3,\mathbb{C})$ be a finite subgroup, let $Y=G\operatorname{-Hilb}(\mathbb{C}^{3})$, and let $\mathcal{R}_{k}$ be the tautological bundle on $Y$ asso…

Equivariant derived McKay conjecture. The derived category of $\mathbb C^{\ast}$-equivariant sheaves on $X$ is equivalent to the derived category of graded finitely generated…

Let $\Gamma$ be a finite subgroup of $SU(n)$ and let $\widetilde{\mathbb{C}^n/\Gamma}$ denote the relevant resolution of the quotient. The representation ring of $\Gamma$ is the ri…

The objects are adjacency graphs associated with fusion rules and modular invariants, together with the graph data arising from discrete subgroups and McKay-type constructions. Hig…

Let $\Gamma$ be a finite subgroup of $SU(n)$, let $M=\mathbb{C}^n/\Gamma$, and let $\widetilde{M}$ be the crepant resolution introduced above. The relevant objects are the cohomolo…