551 problems

For the Simpson stratification of the Hodge moduli space of Higgs/de Rham objects over a smooth complex projective curve of genus $g\geq 2$, Simpson's nestedness conjecture asserts…

For every smooth projective complex variety $X$ and reductive algebraic group $G$, let $h_{\mathrm{sp}}:\mathcal{M}_{G\text{-Higgs}}(X)\to\mathcal{A}_{\mathrm{sp}}(X,G)$ be the spe…

Given a proper morphism $f:\mathcal{X}\to\mathcal{Z}$ with connected fibers between smooth separated Deligne--Mumford stacks, and a smooth effective Cartier divisor…

For every integer $g\ge 2$ and every prime $p$, let $\mathcal{S}_g$ denote the supersingular locus in the moduli space of principally polarized abelian varieties of dimension $g$ o…

Let $\mathcal{C}$ be the moduli space of cubic fourfolds, and let $\mathcal{C}_d$ denote a special divisor, consisting of cubic fourfolds with discriminant $d$. A discriminant $d$…

Higher-dimensional parallelogram decomposition conjecture. Every surface in

Consider the directed graph whose nodes are deformation families of smooth, compact Calabi-Yau threefolds. For two nodes $\mathcal{M}_1$ and $\mathcal{M}_2$, draw an arrow … when,…

Let $f_U:U\to V$ be a smooth projective family, and suppose that $V$ has a morphism $\mu:V\to P_h$, where $P_h$ is the coarse moduli scheme associated with the moduli functor of po…

Let a polarized variety mean a projective variety equipped with a polarization, and consider the moduli space of K-polystable polarized varieties. K-moduli conjecture. A set of K-p…

Let $w^{\alpha}$ be the normal coordinates for the Dubrovin–Zhang hierarchy, let $\tilde u^{\alpha}$ be the normal coordinates for Buryak's double ramification hierarchy, let…

Let $f_{a,b}(z)=z^{3}-3a^{2}z+2a^{3}+b$ be a monic, reduced cubic polynomial with marked critical point $a$, and let $\a7\Sigma_{k,n}$ be the Zariski closure in the moduli space…

Let $n$ be a fixed finite dimension, and consider the moduli space of complex Lie or associative algebras of dimension $n$. A jump deformation is a deformation that is equivalent t…

Let $V_{\mathbb{A}_{1,k}}$ denote the volume of the moduli space of $(1,k)$-annuli, viewed as a polynomial in $\pi^2$. Here $k$ is a positive integer. Leading-term conjecture for a…

Bernoulli-number conjecture. The coefficients satisfy

Persistence Conjecture. If $X$ is arithmetically hyperbolic over $k$, then $X$ is absolutely arithmetically hyperbolic.

Let $X$ be a smooth projective curve of genus $g\geq 2$, let $L\in\operatorname{Pic}^{d}(X)$ with $(2,d)=1$ and $0<d<2$, and let $\mathrm{M}(2,L)$ be the moduli space of stable ran…

Let $X\subset \mathbb{P}^n$ be a general hypersurface of degree $d\leq n$ and dimension at least $3$. Let $R_e(X)$ be the open locus in the Hilbert scheme of $X$ parameterizing smo…

Let $\mathcal{M}_g$ be the coarse moduli space of genus $g$ Riemann surfaces, viewed as a quasiprojective complex variety. Looijenga's conjecture. For every $g\geq 2$, the variety…

Let $\mathcal M_g$ denote the moduli space of closed hyperbolic surfaces of genus $g$, and let $\operatorname{sys}(S)$ be the systole length of $S$. Schmutz Schaller's systole-leng…

Let $(M,J,\omega,\Omega)$ be an almost Calabi–Yau $m$-fold, let $X$ be a compact special Lagrangian $m$-fold in $M$ with conical singularities, and let $\mathcal I_{X'}$ and…

For the Deligne–Mumford moduli space of stable curves $\overline{{\mathcal M}}_{g,h}$, let $\Lambda_g^\vee(u)$ denote the total Chern polynomial of the Hodge bundle and let…

Let $d\geq 2$, let $f:\mathbb{P}^1\to\mathbb{P}^1$ be a degree-$d$ dynamical system, let $h_{\mathcal{M}}$ be a height on the moduli space $\mathcal{M}_d$ of degree-$d$ dynamical s…

Asymptotic normality conjecture. The Betti numbers of these three quotient spaces are asymptotically normally distributed, and the associated variances grow linearly at the same ra…

Tautological generation conjecture. The following assertions hold:

Structural decomposition conjecture.