88 problems

Symplectic duality conjecture. There exist many pairs of symplectic resolutions for which various dualities hold, including Koszul duality between their categories $\mathcal O$.

Let $Y\to X$ and $Y^{\vee}\to X^{\vee}$ be a pair of dual symplectic resolutions with Hamiltonian torus actions. Fix a generic embedding of $\mathbb{C}^{\times}$ into the torus tha…

Kaledin–Lehn–Sorger conjecture. If there exists a symplectic resolution of

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…

Let $\Gamma$ be a finite subgroup acting on $\mathbb C^2$, let $\Gamma_n$ be its wreath product acting on $\mathbb C^{2n}$, and let … be the natural morphism over…

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…

Monodromy conjecture. The monodromy of the quantum connection $\bigtriangledown$ for $X_i$ agrees with the monodromy induced on K-theory by the derived equivalence.

Intersection-cohomology–quantum-cohomology conjecture. The intersection cohomology of $X$ should be isomorphic to the quantum cohomology of $\tilde{X}$ specialized at $q=1$.

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 $\Gamma<\operatorname{SL}(2)$ be a finite subgroup, let $\Gamma_n=\Gamma\wr\mathcal S_n$, and set … A projective or proper symplectic resolution is a projective, respectively p…

Let $\Gamma<\operatorname{SL}(2)$ be a finite subgroup and let … A Mukai flop with center over the origin is a Mukai flop whose flop center is contained in the fiber over $0\in W$.…

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…

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.

Namikawa's conjecture. The variety $W$ is smoothable by flat deformations if and only if it admits a symplectic resolution.

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

Namikawa's conjecture. For any two symplectic resolutions $f_i:X_i\to W$, $i=1,2$, there are deformations

Finiteness conjecture. Any symplectic singularity admits at most finitely many non-isomorphic symplectic resolutions.

Stronger deformation conjecture for symplectic resolutions. There exist 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 $M$ 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…

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…