16 problems

Let $(M,g)$ be a gravitational instanton. Finite-topological-type conjecture. The manifold $M$ has finite topological type, meaning that it can be compactified into a smooth manifo…

Let $(M,g)$ be a gravitational instanton. Asymptotic-cone uniqueness conjecture. The gravitational instanton $(M,g)$ has a unique asymptotic cone. Uniqueness is known for complete…

Let $(M,g)$ be a gravitational instanton, and let an asymptotic cone mean a pointed-Gromov–Hausdorff limit obtained by rescaling $(M,g)$ along a sequence tending to infinity. Metri…

Asymptotic conicity conjecture. A Kähler-Ricci shrinker is weakly asymptotically conical in the sense that it has a unique asymptotic cone at infinity, which is a Kähler cone metri…

Compactification conjecture. For any $\kappa_1>\lambda$ and $\kappa_2>\mu$, $(X,I,\omega,\Omega)$ can be compactified complex analytically to a weak Fano manifold. Furthermore, the…

Quantified asymptotic Euclideanity conjecture. There exists $\delta(N,\epsilon)>0$ such that, for every $\delta\in(0,\delta(N,\epsilon)]$, there exists $R_0\geq 1$ for which…

Let $X_5(\theta)$ be the hyperpolygon space associated with a generic parameter $\theta$. A smooth noncompact hyperkähler manifold is quasi-asymptotically locally Euclidean when it…

Let $X$ be a smooth projective variety of dimension $n$, let $B$ be a line bundle, and for sufficiently large $d$ set $L_d:=\mathscr{O}_X(dA+P)$. Write…

Let $Y$ and $Z$ be unbounded metric spaces, and let $\tilde{r}=(r_n)_{n\in\mathbb{N}}$ be a scaling sequence. For a metric space $W$, write $Seq(W,\tilde{r})$ for its sequence spac…

Let $(X,d)$ be an unbounded metric space and let $\tilde{r}=(r_n)_{n\in\mathbb{N}}$ be a scaling sequence. Define … for $\tilde{x},\tilde{y}\in Seq(X,\tilde{r})$; this is a pseudom…

Gaussian Betti-number conjecture. Fix $q\in[1,n]$. There is a normalizing function $F_q(d)$, depending on $X$ and geometric data, such that for every sequence of integers $p_d$ sat…

Asymptotic vanishing conjecture. Fix $q\in[2,n]$. In the situation of the nonvanishing theorem referenced in the source, there is a constant $C_3$, depending on $X$, $A$, $B$ and…

Let $\mathcal{S}_{nt}^{iso}(\mu)$ denote the isolated-point part of the non-tangential singular set, let $\mathcal{E}$ be the relevant edge set, and let $\partial\mathcal{L}(x)$ de…

Let $\mu$ be a measure under consideration, and let $\mathcal{S}_{nt}^{sing}(\mu)$ denote its non-tangential singular set. Meagreness conjecture. The set…

Let $\mu$ be the measure under consideration, and let $\mathcal{S}_{nt}^{reg}(\mu)$ and $\mathcal{S}_{nt}^{gen}(\mu)$ denote the regular and generic parts of its non-tangential sin…

Let $X$ be a smooth projective variety of dimension $n$, let $L_d$ be the very positive line bundles used to define the Betti numbers $k_{p,q}(X;L_d)$, and write $r_d$ for the rele…