33 problems

Ginzburg's conjecture. The quotient variety $X$ can be naturally desingularized, and this desingularization is holomorphically symplectic and admits a hyperkähler structure. The co…

Let $M$ be a smooth algebraic variety, set $X=T^*M$, and let … be the natural map, assumed to be projective and birational. Demailly–Campana–Peternell conjecture. Then $M=G/P$, whe…

Let $X\to Y$ and $X^+\to Y$ be proper morphisms. They are deformation equivalent if there are deformations $\mathcal X\to\mathcal Y$ and $\mathcal X^+\to\mathcal Y$ over a pointed…

Let $W$ be a symplectic variety, and let $Z\to W$ and $Z^+\to W$ be projective symplectic resolutions. A Mukai flop in codimension $2$ is a birational transformation that becomes a…

Deformation-equivalence conjecture. The resolutions satisfy that $\pi_1$ is deformation equivalent to $\pi_2$, and $Z_1$ is diffeomorphic to $Z_2$.

Finiteness conjecture. The variety $W$ has at most finitely many non-isomorphic symplectic resolutions.

F.-N.'s conjecture. There are flat deformations

Symplectic-resolution deformation-equivalence conjecture. Then $Z_1$ is deformation equivalent to $Z_2$.

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 $V$ 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…

Let $\bar{p}: X \rightarrow \overline{\0}$ be a covering of a nilpotent orbit in a simple Lie algebra $\mathfrak{g}$ such that $R[1]=0$. Suppose that $\pi: Z\rightarrow X$ is a sym…

Let $G$ be a complex connected reductive group, and let $M_{Dol}^{vc}(G)$ be the moduli space of $G$-Higgs bundles with vanishing Chern classes over a compact Riemann surface of ge…

Let $G$ be a connected reductive algebraic group, let $\operatorname{Gr}=G(\mathcal K)/G(\mathcal O)$ be its affine Grassmannian, and let $\lambda_1^\vee,\ldots,\lambda_k^\vee$ be…

Let $\mathfrak{g}$ be a reductive Lie algebra. Choose a well-chosen $\mathbb{Q}$-factorial terminalization … and let $X_{\mathfrak{g},\mathrm{Losev}}$ be the intermediate symplecti…

Let $\root\in\Sig$ be a dimension vector of Type $\II$, and let the associated quiver variety be the variety determined by this dimension vector. Type II uniqueness conjecture. If…

Let $N(\I(\D[4],n))$, $N(\I(\D[m],1))$, $N(\I(\D[m],n)')$, $N(\I(\D[m],0)')$, $N(\I(\E[6,7],n))$, and $N(\I(m))$ denote the numbers of projective crepant resolutions for the indica…

Let $G$ act on ${\bf N}$, let $\nu\colon\mathbb{C}^{\times}\to\mathsf{A}$ define a partial resolution $\widetilde{\mathcal{M}}_{C,\nu}\to\mathcal{M}_C$, and let…

Let $A$ be an alcove in the space of real stabilities, and let $\mathcal{E}(A)$ be the set of joint eigenlines for quantum multiplication by equivariant cohomology classes of $Y$ o…

p-curvature conjecture. The equivariant quantum Steenrod operator satisfies

Let ${\mathfrak{X}}$ be the symplectic-resolution object under consideration, and let $\Xi_{{\mathfrak{X}}}$ be the parameter set for elliptic canonical-basis classes. Let Properti…

Let $\fg$ be an orthogonal Lie algebra of type $B$ or $D$, of rank $n$, and write a Levi subalgebra as … with $b_1\geqslant\cdots\geqslant b_k\geqslant1$. Let…

$\mathcal{O}$-singular locus conjecture. The locus $\mathfrak{c}^{\mathcal{O}-\mathrm{sing}}$ coincides with the union of singular hyperplanes.

Let $X$ be a compact Riemann surface with marked points and fixed parabolic parameters, and let the corresponding character variety parametrize representations of the fundamental g…

Let $X$ be a conical Poisson variety with a symplectic resolution, and let $d$ be the degree of its generic symplectic form. Let $\widetilde X$ be the resolution and, for each $i$,…

Let $\rho:\widetilde X\to X$ be a symplectic resolution, with $X$ affine, and let $\operatorname{\mathsf{HP}}_i^{DR}(X)$ denote the Poisson-de Rham homology of $X$. The affine symp…