193 problems

Let $(V,0)$ be a normal surface singularity in $\mathbb C^4$ that is not a complete intersection. Milnor–Tjurina conjecture in C4. … with equality if and only if $(V,0)$ is quasi-h…

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…

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

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…

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

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…

KGB obstruction conjecture. The KGB obstruction is the only obstruction to the local lifting problem for local $G$-extensions where $G$ has cyclic $p$-Sylow subgroup. In particular…

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…

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 $D\in |{-}K_X|$ when such an ele…

Variational Hodge conjecture. The following are equivalent:

Defect bound conjecture. We have

Generic dimension and non-smoothability conjecture. For generic $L_r^n$, equality holds in the stated bounds for $\dim T^1_l$ when $l>0$ and for $\dim T^1_{-1}$, with some exceptio…

Let $E$subset W be the special fiber consisting of a nodal curve on an algebraic surface, let $\mathop{\operatorname{Def}}_{(E \subset W)}$ be the versal deformation space of the p…

Let $A$ be a geometric $n$-stack, and let $D=\operatorname{Spec}(k[\epsilon]/(\epsilon^2))$. Let $TDef_A$ denote the $(n+1)$-category of deformations of $A$ parametrized by $D$. Le…

Let $p$, the finite field $k$, the deformation data $\mathcal D$, the absolutely irreducible type-$\mathcal D$ representation $\bar\rho:G_{\bold Q}\to\operatorname{GL}_2(k)$, and t…

Let $p$ be an odd prime, let $k$ be a finite field of characteristic $p$, and let deformation data be a pair $\mathcal D=(\Sigma,t)$ consisting of a finite set of primes and one of…

Kleppe–Ellia's conjecture. If $C$ is linearly normal, then every small deformation $C'$ of $C$ in $\mathbb P^3$ is contained in a cubic surface $S'\subset\mathbb P^3$; equivalently…

Formality conjecture. The Hochschild complex of the algebra $C^\infty(M)\rtimes G$ is a formal differential graded Lie algebra.

Let $R_{\rho,\mathcal F}$ be the ring pro-representing the refined deformation functor, let $\mathbb T$ be the completed local Hecke algebra at the corresponding eigenvariety point…

Let $E$ be the number field in the deformation problem, let $V$ be an absolutely irreducible crystalline $p$-adic representation with the stated self-duality and distinct local dat…

Let $X$ be a proper rigid analytic space over $K$. A formal versal deformation of $X$ is the formal deformation supplied by the local deformation theory of $X$. Global versal-famil…

Let $X\to Y$ and $X^+\to Y$ be proper morphisms. They are deformation equivalent if there are deformations $\mathcal X\to\mathcal Y$ and $\mathcal X^+\to\mathcal Y$ over a pointed…

Deformation-equivalence conjecture. The resolutions satisfy that $\pi_1$ is deformation equivalent to $\pi_2$, and $Z_1$ is diffeomorphic to $Z_2$.

Let $A$ be a ring and let … be a bounded complex of finitely generated $A$-modules. Realization conjecture. Every such complex arises as the direct image complex of a family of she…

Radicality conjecture. The ideal above is radical. This stronger assertion includes cases involving certain unions of splitting-type strata. The source records the case $r=2$ as as…