22 problems

Let $(X,B)$ be a pair, and let $f\colon X\to S$ be a contraction such that $-(K_X+B)$ is nef over $S$. For an arbitrary schematic point $s\in S$, let $f^{-1}(s)$ denote the scheme-…

Ducat's conjecture. Then the pair $(X,B)$ is log rational. The conjecture extends the demonstrated surface and projective three-space examples, where Cremona transformations relate…

Anticanonical dual-complex conjecture. The dual simplicial complex of an anticanonical simple normal crossings divisor should be a sphere, or a closely related space.

A log Calabi–Yau variety is a smooth complex quasi-projective variety admitting a log compactification whose boundary is a simple normal crossing divisor and whose logarithmic cano…

Let $p$ be a prime and let $k$ be an integer. Consider a dlt log-Calabi–Yau $4$-fold pair, meaning a dlt pair of dimension $4$ whose log canonical divisor is numerically trivial, a…

Local closedness and finiteness conjectures. The locus $\mathcal{D}^{\rm KV}(X,\Delta+D)$ satisfies all of the following:

Let $(X,B)$ be a log Calabi--Yau pair of dimension $n$. Write $4\mathcal{D}(X,B)4$ for its dual complex, and let $c_{\rm bir}(X,B)$ denote its birational complexity. Let $\Sigma^n$…

Let $(X,B)$ be an $n$-dimensional log Calabi–Yau pair, meaning that $X$ is a proper variety, $B$ is an effective $\mathbb{R}$-divisor, $(X,B)$ is log canonical, and $K_X+B$ is…

Let $(X,B)$ be a log Calabi–Yau pair, meaning that $X$ is a proper variety, $B$ is an effective $\mathbb{R}$-divisor, $(X,B)$ is log canonical, and $K_X+B$ is $\mathbb{R}$-linearly…

Let $f\colon Y\to S$ be a special Lagrangian torus fibration of a Calabi–Yau variety $Y$, with base $S$. Kontevich–Soibelman conjecture. The space $S$ should be a manifold or, at l…

Virtually nilpotent kernel conjecture. The normal subgroup $N$ is virtually nilpotent of rank at most $2n+1$.

Let $X$ be an $n$-dimensional Fano type variety of coregularity zero. A $2$-complement $(X,B)$ is a complement with $(X,B)$ log canonical and $2(K_X+B)\sim 0$. Let $(Y,B_Y)$ be the…

Let $(X,B)$ be an $n$-dimensional log Calabi–Yau pair, and let $\mathcal{D}(X,B)$ denote its dual complex. A finite cover $\pi\colon Y\rightarrow X$ is equipped with the log pull-b…

Let $S$ be a simply-connected surface with Hodge numbers $h^{1,0}=h^{2,0}=0$. A geometrically maximal degeneration is a degeneration having the geometric maximality property used i…

Let $\mathscr{X}$ be a maximally unipotent good minimal dlt degeneration of compact hyper-Kähler manifolds over $\mathbf{C}((t))$. Let $\mathcal{D}(\mathscr{X}_0)$ denote the dual…

Let $M_B$ be the Betti moduli space, and let $\mathcal{D}(\partial M_B)$ denote the dual complex associated with the boundary of a dlt compactification. Dlt log Calabi--Yau compact…

Let $M_B$ and $M_{\mathrm{Dol}}$ be the Betti and Dolbeault moduli spaces, respectively. Let $N^*_B$ and $N^*_{\mathrm{Dol}}$ be punctured neighbourhoods at infinity, let…

Minimal skeleton conjecture. The homeomorphism type of the dual complex of $(\mathcal{X},\sum_{1\le i\le s}\mathcal{D}_{i})$ does not depend on the choice of such compactification.

Let ${\rm M}_{Dol}(S,C_{\cdot})$ be the Dolbeault (Hitchin) moduli space and ${\rm M}_B(S,C_{\cdot})$ the Betti character variety. Let $N^{\ast}_{Dol}$ and $N^{\ast}_B$ be neighbor…

Let $(X,\Delta)$ be a log Calabi--Yau pair, meaning that $X$ is a proper variety and $K_X+\Delta$ is numerically trivial. Write $X^{\rm sm}$ for the smooth locus of $X$ and let…

Let $D$ be the exceptional divisor of a resolution of an isolated rational hypersurface singularity of dimension at least $3$, and let $\Delta=\Delta(D)$ be its dual complex. Ratio…

Let $f:X\to Y$ be a good resolution of an isolated rational singularity $o\in Y$, and let the exceptional divisor be … with effective simple normal crossings support. Let $\Delta$…