20 problems

Unirationality conjecture. Unirationality characterizes supersingularity for symplectic varieties.

Generalized Bogomolov decomposition conjecture. There is a finite étale cover $Y'\to Y$ such that $Y'$ is birationally equivalent to

Exceptional-locus conjecture. The exceptional locus is isomorphic to

SYZ Conjecture. The line bundle $L$ is semiample.

Finiteness conjecture. The fundamental group $\pi_1(M_\mathrm{reg})$ is finite.

A primitive symplectic variety is a normal compact Kähler variety with symplectic singularities whose smooth locus carries a symplectic form and whose reflexive powers of the canon…

Finiteness conjecture. The group $\pi_1(X^{\mathrm{reg}})$ is finite:

SYZ conjecture. Every such $X$ admits a locally trivial deformation to a primitive symplectic variety with a Lagrangian fibration.

Generalized abundance for symplectic varieties. Then $L$ is semiample.

Second-Betti-number bound. One should have

Riemann--Roch root conjecture. 1. The polynomial $\mathrm{RR}_X(q)$ has $n$ distinct negative real roots forming an arithmetic sequence. 2. If $X$ is smooth, then…

Supersingularity conjecture. If $X$ is second-Artin supersingular, then:

$3d$-mirror symmetry. For every symplectic variety from this class, there exists a dual variety $X'$ called the $3d$-mirror of $X$.

Section property conjecture. This algebra epimorphism admits a multiplicative section whose image contains all Chern classes of the tangent bundle of $X$.

Geometric characterization conjecture. Second Artin supersingularity is equivalent to unirationality, and it is also equivalent to rational chain connectedness.

Decomposition conjecture. There exists a quasi-étale cover $Y \to X$ such that either $Y$ is an abelian variety, or $Y$ splits as a product of copies of a single Calabi–Yau variety…

Primitive symplectic decomposition conjecture. Primitive symplectic varieties should be building blocks in a singular version of the Beauville–Bogomolov decomposition theorem. This…

Let $M_{A,2d}$ be the modular variety associated with the lattice and polarisation considered in the source, with $d=2^n$. Non-general-type conjecture. The variety $M_{A,2d}$ is no…

Let $\mathcal M_h$ be the moduli space of $10$-dimensional polarised O'Grady varieties with a non-split polarisation $h$, and let $2d=h^2$ be its Beauville degree. Unirationality c…

Let $L=3U\oplus 2E_8(-1)\oplus A_2(-1)$ be the Beauville lattice of an O'Grady $10$-dimensional variety, let $h\in L$ be a polarisation, let $\mathcal M_h^0$ be a component of its…