16 problems

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…

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…

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 $M_B$ 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…

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 $D$ 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 $\Delta$…