193 problems

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…

Let $M$ be a symplectic $C^\infty$-manifold or an affine complex algebraic variety, let $G$ be a finite group acting on $M$ by symplectic transformations, and let $B$ be a $G$-equi…

Let $B$ be a local commutative algebra over $k$ with maximal ideal ${\mathfrak m} \subset B$, and let $\widetilde{A}$ be a flat associative algebra over $B$ such that…

cDV smoothing conjecture. $V$ has a smoothing by a small deformation. In particular,

Let $A$ be a Lie algebroid that admits a proper integrating groupoid $\mathcal{G}$ whose $s$-fibers are $2$-simply connected. Cohomological rigidity conjecture. Then … This conject…

Symplectic-resolution deformation-equivalence conjecture. Then $Z_1$ is deformation equivalent to $Z_2$.

Let $K$ be a finitely generated free ${\cal D}_X$-module, and let $SK$ denote the associated chiral algebra. Let $\mathrm{def}_{SK}$ be its deformation pro-$*$-Lie algebra, and let…

Selmer cotangent-space conjecture. The inclusion is an isomorphism:

Let $S$ be the fixed set of primes and let $S'=s'_1,\dots,s'_n\}$ be a ramified auxiliary set whose existence is guaranteed by Theorem 1. The ring $R_{S\cup S'}$ is a finite flat c…

Let $X\in\mathcal C$ be smooth, and let $\operatorname{ess}(X)$ denote the specialness invariant introduced in the paper. Deformation conjecture. The invariant…

Homotopical constancy conjecture. If the isolated singularity $(X_0,f_0)$ does not split, then the Milnor fibration of the isolated singularity of $(X_t,f_t)$ is homotopically cons…

Bouquet conjecture. The Milnor fibre of $(X_0,f_0)$ is homotopy equivalent to the bouquet of the Milnor fibres of the isolated singularities into which it splits.