42 problems

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 $\pi:\mathcal X\to\Delta$ 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 $X_t$ 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 $\mu:Y\to X$ 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…

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…

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…

Let $\pi:\mathcal{X}\to\Delta$ be a semistable degeneration, with smooth projective fibers $X_t$ for $t\neq 0$, singular central fiber $X_0$, smooth total space, and monodromy loga…

Nagai's conjecture. The monodromy operators satisfy

Consider a family of unitary conformal field theories with fixed central charge $c$, parameterized by $\varepsilon\to 0$, and closed surfaces $\Sigma$ in the family described by lo…

Let $X\to \mathbb{D}^*_t$ be a polarised meromorphic degeneration of $n$-dimensional Calabi–Yau manifolds, with relatively ample line bundle $L$, and let $X_K^{an}$ be the associat…

Let $\mathfrak{X}\to\Delta$ be a flat, projective degeneration with reduced special fiber $X_0$, and let $\pi\colon\mathfrak{X}^*\to\Delta^*$ be smooth, where…

Let $R$ be a DVR with fraction field $K$ and residue field $k$. Let $X$ be a flat, projective $R$-scheme such that the generic fiber $X_K$ is smooth and separably rationally connec…

Let $(X,\Delta)$ be a log Fano pair and let $g,\bar{g}$ be functions satisfying the conditions in the source. Let $(Y,\Delta_Y,\xi_0)$ and…

Let $(\mathcal{X},\mathcal{L})\to\Delta$ be a non-log-terminal semi-log-canonical, locally stable degeneration of polarized log-terminal projective Calabi--Yau varieties, and enhan…

Let $\pi\colon\mathcal{X}\to\Delta^{\rm conv}(\mathbb{C})$ be an affine finite-type log-terminal faithfully flat morphism, where…

Let $f:X\longrightarrow C$ be a proper semi-stable degeneration over a smooth curve $C/k$, with $k\subset\mathbb C$. Let $X_t$ be a fiber at a complex point $t\neq s$, and write it…

Nonexistence conjecture. For $n=6$, one has

Let $X$ be a Calabi–Yau variety, or let $X\setminus D$ be a log Calabi–Yau variety. Suppose that $X$ (or $X\setminus D$) admits a semi-stable degeneration connected to a point of m…

Let $X$ be a quasi-Fano variety with a smooth anticanonical divisor $D$, and let $D_1+D_2$ be a simple normal crossings anticanonical divisor of $X$. Let $(X^\beth,h)$ denote the r…

Let $X$ be an $n$-dimensional Calabi–Yau manifold undergoing a Type $N$ degeneration $\mathcal{X}\to\Delta$, where Type $N$ means that the logarithm of monodromy has nilpotency ind…

Conjecture on the quadratic conductor formula. Then