28 problems

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 $\mathcal{G}_{r}$ be the nontrivial $\mu_{r}$-gerbe over $\mathbb{P}^{d}$ and let $\mathbb{P}^{d}\times\mathcal{B}\mu_{r}$ be the trivial $\mu_{r}$-gerbe over $\mathbb{P}^{d}$.…

Let $\mathcal{G}_{r}=\mathbb{P}(r,\ldots,r)$ be the canonical nontrivial $\mu_{r}$-gerbe over $\mathbb{P}^{d}$, let $H$ denote the hyperplane class of $\mathbb{P}^{d}$, and let $q$…

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…

Ginzburg–Kaledin's conjecture. The isomorphism is multiplicative and extends from symmetric powers to any symplectic orbifold.

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 $M$ 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…

The Chen–Ruan orbifold cohomology ring assigns an obstruction bundle to each relevant sector of a complex orbifold. Riemann-bilinear-relations conjecture. The Riemann bilinear rela…

Let $\mathcal{X}$ be a smooth Deligne--Mumford stack with an étale cover $\pi:U\to\mathcal{X}$ by a scheme, and set $Z=U\times_{\mathcal{X}}U$. The correspondence $Z\times_UZ\to Z$…

Let $X$ be a quasiprojective scheme over a field $k$, let $G$ be a finite group acting on $X$, and assume that $\operatorname{char} k$ does not divide $|G|$. Let…

Ruan's quantum-corrected conjecture. There exists a ring isomorphism between

Chen–Ruan's conjecture. The ring $H_{CR}^*(X)$ coincides with the cohomology ring $H^*(S^{[n]})$ of the Hilbert scheme of $n$ points on $S$.

D-orbifold string theory conjecture. There is an inner local system $\mathcal L$ such that the ordinary cohomology of $Y$ is the direct sum of $H^*_{orb}(X,\mathcal L)$ and the coh…

Let $X$ be a Calabi–Yau Gorenstein orbifold, and let a desingularization of $X$ be given. An inner local system is a compatible assignment of a flat orbifold line bundle to each se…

Crepant Resolution Conjecture. There exists a graded linear isomorphism of rings

Wang's maximal-Betti-number conjecture. The variety $W^{(n)}$ has the largest sum of Betti numbers among smooth, projective Calabi-Yau $n$-folds. When $n$ is odd, it also has the s…

Let $n$ be a positive integer, and consider projective $n$-dimensional varieties with quotient singularities and trivial canonical class. Write $H$ for the sum of their orbifold Be…

Todd-class-corrected HKR compatibility conjecture. The diagram

Bigrading-preservation conjecture. In Calabi–Yau situations, the two products agree; equivalently, every component map in the original product that does not preserve the bigrading…

Let $\mathscr{X}$ be a projective symplectic quotient orbifold, and write $H\! H^\bullet(\mathscr{X})$ for its Hochschild cohomology and $H^\bullet_\mathrm{orb}(\mathscr{X})$ for i…

Let $\mathscr{X}$ be a symplectic quotient orbifold and let $\mathscr{A}$ be an appropriate quantization of $\mathscr{X}$. Ginzburg–Kaledin conjecture. There is an algebra isomorph…

Let $X$ be a projective symplectic variety over $\k$ and let $G$ be a finite group acting on $X$ by symplectic automorphisms. Write $[X/G]$ for the quotient orbifold, and let…

Lift-independence conjecture. The resulting correlators are independent of the choice of lift.

Let $M$ be a smooth projective holomorphic symplectic variety equipped with a faithful action of a finite group $G$ by symplectic automorphisms of $M$. If $Y$ is a symplectic resol…

Let $(W,G)$ be of Calabi–Yau type: $W$ is nondegenerate, not necessarily invertible, with weights $q_j$ satisfying … and $G$ contains $\langle j_W\rangle$ and lies in $SL_W$. Set…