43 problems

Let $X$ be an equal-rank locally symmetric space, let $X^*$ be a Satake compactification, and let $\widehat{X}$ be Zucker's reductive Borel–Serre compactification. Let … be the quo…

The GIT compactification and the Baily–Borel compactification of the moduli space of quartic K3 surfaces are given as two ends of an interpolation by the HKL model, the model defin…

Let $X^{\circ}$ be the space of nondegenerate configurations of $n+1$ hyperplanes in ${\mathbb P}^{n}$ defining tetrahedra, let $X$ be its canonical compactification, and let…

Simon's conjecture. If there is no sequence in $\mathbb{N}$ converging to a point of $b\mathbb{N}\setminus\mathbb{N}$, then there is a continuous map

Let $K$ be a countable CW complex, and let $AE(X)$ mean that every map from a closed subset of $X$ to $K$ extends over $X$. A metrizable compactification $\widetilde{X}$ is a compa…

Let $V$ be a variety with only rational singularities. Compactification conjecture. The variety $V$ admits a compactification with only rational singularities. The source gives no…

Let $Y$ be a Calabi–Yau manifold with $h^{2,0}(Y)=0$, let $\pi:\mathcal{Y}\to S$ be a family of complex structures on $Y$ with everywhere-isomorphic Kodaira–Spencer map, and let…

Arithmetic quotients of bounded symmetric domains admit analogues of Satake–Baily–Borel compactifications with a kind of minimality property among compactifications. Griffiths's co…

Let $\mathcal{X}(k,n)$ be the open configuration space of $n$ labeled points in $\mathbb{P}^{k-1}$ in general position, and consider its log canonical compactification and the norm…

Let $D$ be the boundary divisor in a compactification of the relevant character variety, and let the dual intersection complex of $D$ be the polyhedral complex encoding the interse…

Let $G$ be a reductive Lie group, let $g$ be the genus of a closed Riemann surface, and let $\mathcal{X}_{g}(G)$ denote the corresponding character variety. A dlt log Calabi–Yau co…

Milne's compactification conjecture. Smooth integral models of Shimura varieties of abelian type should have toroidal and minimal compactifications with the same properties mention…

Let $X$ be a smooth compactification of $\mathbb{C}^n$ with $b_2(X)=1$, and suppose that $X$ is Kähler and its divisor at infinity is smooth. Folklore conjecture. Then $X$ must be…

Let $X$ be a compact complex manifold of complex dimension $n$, and let $Y$ be a subvariety such that $X\setminus Y$ is biholomorphic to $\mathbb{C}^n$. The pair $(X,Y)$ is standar…

Griffiths conjecture. The period image $\mathlcal P$ is quasi-projective and $\varphi$ is induced by an algebraic morphism; $L$ is ample on $\mathlcal P$; and there exists a Baily–…

Relative compactification conjecture. For some $n_0\in{\mathbf N}$, $P_{n_0}$ is a relative compactification of ${\mathcal G}$ and

Let $S$ be a parameter space with compactification $\overline{S}$, let $\Phi:S\to\wp$ be a period map with image $\wp=\Phi(S)$, and let $\Phi^*:\overline{S}\to\wp^*$ be a proper, c…

Let $D$ be a period domain and let $\Gamma$ be a discrete subgroup acting on $D$. A horizontal completion of the period map is a compactification extending the period map to the bo…

Shokurov–Odaka–Laza conjecture. There exists a compactification of the moduli space of smooth Calabi–Yau varieties on which the Hodge line bundle is ample and the boundary parametr…

Locality conjecture. There exists a $G(\sF)\times G(\sF)$-equivariant sheaf ${\mathbf S}_{\rho}$ of complex vector spaces on $\overline{G}(\sF)$ such that…

A weak K-moduli space is an Artin stack when continuous isotropy is allowed. Consider such a stack parametrizing polarized irreducible holomorphic symplectic varieties, polarized a…

Let $\overline{\mathcal{Y}}_{n,k}$ and $\overline{\mathcal{Z}}_{n,k}$ be the compactifications of $\mathcal{Y}_{n,k}$ and $\mathcal{Z}_{n,k}$ described in the source, let…

Let $g:X\to Y$ be a smooth map of smooth varieties. Consider the relative Fulton--MacPherson spaces associated with $g$ and the corresponding relative version of the Ulyanov spaces…

Snug embedding conjecture. Any suitable compactification $q$ admits a snugly embedded trunc $G$ in

Let $G$ be a Lie group, $K$ a maximal compact subgroup, $X=G/K$, and $S=\Gamma\backslash X$ an arithmetic locally symmetric space. Let $\overline{\Gamma\backslash X}^{S}_{\rm max}$…