78 problems

Nagai's conjecture. The monodromy operators satisfy

Grunewald–O'Halloran conjecture. For every $n$-dimensional nilpotent Lie algebra $\mathcal A$, there exists an $n$-dimensional non-nilpotent Lie algebra $\mathcal B$ such that

A smooth degeneration of a projective manifold is a smooth proper family whose general fibre is a projective manifold. A compact complex manifold is Moishezon if its algebraic dime…

Vertex degeneration conjecture. For any vertex $v$ in the generic marked poset polytope, there exists a vertex $t_v$ of the hypercube $[0,1]^{\tilde P}$ such that the image of $v$…

Let $X$ be a rational surface and let $\mathcal{A}$ be a toric system on $X$. Denote by $\mathbb{TV}(\mathcal{A})$ the smooth toric variety associated with $\mathcal{A}$. Bondal's…

Vakil's geometric Littlewood–Richardson conjecture. There exists a subset $U\subseteq S\times S\times\{d_0,\dots,d_{\binom n2}\}$ such that: $(\alpha,\beta,d_0)\in U$ for all…

Let $Fl(n)$ be the flag variety, let $d_i$ and $d_{i+1}$ be consecutive elements in the specified degeneration order, and let an equivariant puzzle variety be the subvariety of…

Let $\pi:\mathcal X\to\Delta$ be a degeneration of an irreducible symplectic $2n$-fold. For each $m$, let $T_m$ be the associated monodromy operator on…

Let $\pi:\mathcal X\to\Delta$ be a degeneration of an irreducible symplectic $2n$-fold. For each $m$, let $T_m$ be the associated monodromy operator on…

Let $X_0$ be a normal crossing variety with smoothing $X_t$, and let $X$ denote the associated total space. Write $H^k(-,\mathbb Z)_f$ for the torsion-free quotient of integral coh…

Let $M=M_{\leq}+M_{>}$ be decomposed into its lower-triangular and strictly upper-triangular parts, and define … Let $D_0$ be the flat limit as $t\to0$, and let $M\bullet M$ denote…

Let $\mathcal V\to\Delta^*$ be a variation of mixed Hodge structure arising from a family of complex algebraic varieties, where $\Delta^*$ is a punctured disk. A period map is said…

Vermutung. Every immobile curve $X$ of genus $g\geq 2$ satisfies

Smooth-limit hypersurface conjecture. Every such smooth limit is again a hypersurface.

Let $X_t$ be a smooth family of compact holomorphically symplectic manifolds over a disk $\Delta$. Assume that $X_t$ is hyperkähler for all $t\neq 0$. Soldatenkov–Verbitsky conject…

Let $X_t$ be a degeneration limit of hyperkähler manifolds. A degeneration limit of hyperkähler manifolds should belong to Fujiki class $\mathcal C$. This conjecture concerns wheth…

Finite-energy non-Archimedean equality conjecture. Equality is obtained provided one includes finite-energy non-Archimedean metrics.

Let $\mu:Y\to X$ be the birational modification in a semistable reduction, let $f:Y\to T$ be the induced family, let $F^{*}\xi$ be the pulled-back coefficient bundle, and let…

Let $\pi:X\to S$ be the degeneration under consideration, with central fibre $X_{0}$, relative canonical bundle $K_{X/S}$, and let $H$ be an ample line bundle on $X$. Write…

Flat-locus dimension conjecture. A tuple $f\in R^{\iota}_{\bf d}$ is in the flat locus of $\pi$ if and only if

Type A minimal-dimension conjecture. There exists a representation $M^2$ of dimension $\dim P$ such that $\Gr_{\dim P}(N)$, for $N\in{\rm Rep}_{\bf d}$, has dimension…

Irreducible-locus conjecture. There exists a representation $M^1$ of dimension $\dim P$ such that, for $N\in{\rm Rep}_{\bf d}$, the quiver Grassmannian $\Gr_{\dim P}(N)$ is irreduc…

Higher-dimensional eigenvalue degeneration conjecture. The following assertions hold: (1) for every $k\in\mathbf N$, $\lambda_k(s)$ extends continuously to $S$; (2), with

Let $Y_t$ be a smooth fibre of a semistable degeneration with nodal central fibre $Y_0$, let $\beta\in H^{2,2}(Y_t,\mathbb{Q})$ be rational, and let…

Let $\pi:\mathcal{Y}\to\Delta$ be a $G$-equivariant semistable degeneration of Calabi–Yau threefolds with ordinary double points, and let $\widetilde{Y}_0$ be a $G$-equivariant cre…