43 problems

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 $S$ 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…

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

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 $V$ be a vector space, let $G\subset SL(V)$ be a finite subgroup, and let $X\to V/G$ be a crepant resolution. Write $D^b_G(V)$ for the bounded derived category of $G$-equivaria…

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…

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…

Physicists' Euler number conjecture. In appropriate circumstances,

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 $G\subset GL_2({\mathbb{C}})$ be any finite subgroup, and let $X$ be a resolution of $\mathbb{C}^2/G$. Let $Y$ be the maximal resolution of the pair $(\mathbb{C}^2/G,B)$, where…

Exceptional-curve conjecture. $\operatorname{EC}(a+b+c)$ corresponds to the exceptional curves of $G\text{-}\operatorname{Hilb}(\mathbb{C}^{3})$.

Let $X$ be the smooth Calabi–Yau complete intersection whose existence is conjectured to match the $A$-periods and GKZ system of $Y_{\mathrm{nc}}$, and let $X_{\mathrm{pre}}$ be th…

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $G\subseteq \mathrm{SL}(n,k)$ be a finite small group of finite representation type, meaning that there ar…

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 $X$ 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 $Y$…

Let $S$ be a rational homology $\mathbb{CP}^2$ with at most $4$ cyclic singularities, and let $S^0$ denote its smooth locus. Reduced Algebraic Montgomery–Yang conjecture. If $S^0$…

PIA conjecture. One has

Let $G\subset GL(V)$ be a finite group, let $F$ be a field, and let $V$ be a finite-dimensional $F$-vector space. Let $S(V)$ be the symmetric algebra of $V$, and write $S(V)^G$ for…

Shokurov's Gorensteinness conjecture. The following assertions hold:

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 $r(n,a)$.