402 problems

Let $A$ be an algebra, $M$ an $A$-module, and let the deformation complex be the controlling differential graded Lie algebra associated with the deformation problem of $M$, such as…

Let $f:X\to S$ be a smooth family from a Kähler manifold $X$ onto a smooth, connected, relatively compact curve $S$. Assume that $(X,B+\boldsymbol{\beta})$ is a generalized klt pai…

Grunewald–O'Halloran conjecture. Every complex finite-dimensional nilpotent Lie algebra of dimension greater than two is the degeneration of another Lie algebra.

Peternell's conjecture. Any minimal Kähler variety has an algebraic approximation.

Kaledin–Lehn formality conjecture. The differential graded algebra $\operatorname{RHom}(\mathcal{F},\mathcal{F})$ is formal, namely, it is quasi-isomorphic to its cohomology algebr…

Donald–Flanigan conjecture. The group algebra $kG$ admits a $k((t))$-separable deformation.

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…

Fujiki class conjecture. The central fiber $\mathcal{X}_0$ is a manifold in the Fujiki class $\mathcal{C}$.

Let $X_{0}$ be a Calabi–Yau threefold over an algebraically closed field $k$ of characteristic $p$. Write $h^{1,0}$ for its Hodge number and $h_{W}^{1,2}$ for the corresponding cry…

Bloch's semiregularity conjecture. The map $\tau$ annihilates all obstructions, giving a reduced obstruction space.

Let $X$ be a Q-Fano threefold, meaning a three-dimensional Fano variety with $Q$-Cartier canonical divisor. Let $Delta^1$ be a unit disc, and let $Din |{-}KX|$ when such an elemen…

Let $X\to B$ and $X^\phi\to B$ be projective Lagrangian fibrations related by a Tate-Shafarevich twist, where a Tate-Shafarevich twist is obtained by modifying the transition funct…

Let $f:\mathscr{X}\to\mathbb{D}$ be a smooth projective family over the unit disk, and write $X_t=f^{-1}(t)$. A fiber has big fundamental group when the image of the fundamental gr…

Let a finite quotient singularity be a singularity obtained as a quotient by a finite group action. Riemenschneider's conjecture. Finite quotient singularities should deform to fin…

Let $p$ be a prime, $K$ a number field, $S$ a finite set of primes of $K$, and $\overline{\rho}:G_{K,S}\to \operatorname{GL}_{n}(\mathbb{F})$ a continuous absolutely irreducible re…

Let $\mathscr O_\mathrm{def}(\mathbb B^n_r)$ denote the deformed algebra of holomorphic functions on the ball $\mathbb B^n_r$. Theorem establishes that the analogous deformed algeb…

Let $F$ be a field, let $\Gamma_F$ be its absolute Galois group, and let $k$ be a field. A finite-dimensional complete flag of continuous representations of $\Gamma_F$ over $k$ is…

Gouvea's dimension conjecture. In the presentation proposition for the universal deformation ring, equality always holds in the Krull-dimension bound; equivalently,

Let $(X,0)$ be a rational surface singularity, and consider its resolution. Assume that every exceptional curve in the resolution has self-intersection at most $-5$, and let the ba…

Let $R$ be the deformation ring, let $\overline R$ be the corresponding deformation ring after replacing $e$ by $e+1$, and let $O^{\nu,\Delta}_{\mu,R}$ and…

Mazur's dimension conjecture. The ring $R$ is flat over $W(k)$ and its relative dimension over $W(k)$ is $\delta(\bar{\rho})$. The conjecture predicts that the expected dimension f…

Invariance of plurigenera conjecture. For every $m$, the quantity $P_m(X_t)$ is independent of $t\in B$. The claim predicts deformation invariance of these dimensions in smooth fam…

Let ${\cal S}$ denote the category constructed in the paper from the elliptic spherical double affine Hecke algebra, and let $X_m$ be the corresponding surface. Flat-deformation co…

Let $(X,E)\rightarrow (V,0)$ be the minimal good resolution (MGR) of a complex normal surface singularity, and define $S_X=(\Omega^1_X(\operatorname{log} E))^*$. Assume that…

Flexibility conjecture. These types of real projective orbifolds with R-ends might be very flexible.