48 problems

Shokurov's toric-model conjecture. Every maximal log Calabi–Yau pair $(X,\Delta_X)$ whose underlying variety $X$ is a rational $3$-fold has a toric model.

Let $X$ be a normal projective variety, let $f\colon X\to X$ be a $q$-polarized endomorphism, and let $\Delta$ be an effective $\mathbb{Q}$-divisor such that $(X,\Delta)$ is an $f$…

Stable-pair/Donaldson–Thomas correspondence conjecture.

Let $\bold X$ be a logarithmic Calabi–Yau 3-fold, let $D^{[\beta]}$ be the product of the relevant Hilbert schemes of points on the interiors of the boundary divisors, and let … be…

Let $(X,D)$ be a log Calabi–Yau manifold, with $D$ an anticanonical divisor of $X$, and let $W_X\colon \check X\to\mathbb C$ be a mirror of $(X,D)$. Let $\widehat{\Lambda}_{\geq}$…

Let $(Y,D)$ and $(Y',D')$ be $n$-dimensional, deformation-equivalent log Calabi–Yau pairs, where a log Calabi–Yau pair is a smooth projective variety together with an anticanonical…

Let $U\subset Y$ be an snc partial minimal model, let $W$ be an effective ample divisor supported on the boundary, and let $\widetilde A_{\theta}$ be the graded subring obtained fr…

Let $U\subset Y$ be a smooth affine partial minimal model with smooth boundary $E$, let $V$ be the mirror fibre, and let $\theta_E$ be the corresponding theta function. Let $F$ be…

Let $U$ and $V$ be affine log Calabi–Yau varieties with maximal boundary, and let $\operatorname{Sk}(U)$ and $\operatorname{Sk}(V)$ be their skeleta. For $u\in\operatorname{Sk}(U)$…

Let $U$ be an affine log Calabi–Yau variety and let $V$ be a general fibre of its mirror family; assume the mirror construction extends to the canonical-singularity setting conside…

Let $(X,B)$ be a log canonical log Calabi–Yau pair, meaning that $K_X+B$ is numerically trivial, such that $\dim X=3$, $\operatorname{coreg}(X,D)=0$, and $X$ is rational. Ducat's c…

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…

Keel–Yu mirror detropicalization conjecture. (1) In the presence of multiple open tori, the integral points of $\mathrm{sk}(U)$ can be equipped with the structure of a polyptych la…

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…

For a positive integer $n$, let $\mathcal{F}_n$ be the family of $n$-dimensional projective varieties $X$ such that there exists a log Calabi--Yau pair $(X,B)$ with … and…

Let $\mathcal{T}(X,B)$ denote the torus exceptional degree of a log Calabi--Yau pair $(X,B)$. Torus exceptional degree boundedness conjecture. For every positive integer $n$, there…

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 $n$ be a positive integer, and let $(X,\Delta)$ be an $n$-dimensional log canonical log Calabi–Yau pair. Write $\pi_1^{\rm orb}(X,\Delta)$ for its orbifold fundamental group. S…

Semiampleness conjecture. The generalized Hodge line bundle $\lambda$ is semiample.

Let $X$ be a log Calabi–Yau variety, and let $\mathbb{D}\partial X$ denote its dual boundary complex. Kontsevich's conjecture. The dual boundary complex $\mathbb{D}\partial X$ is h…

Mirror-symmetry compactification conjecture. For every $n>3$ and generic $\boldsymbol{t}\in\mathbb{C}^n$, there exists a smooth compactification…

Generalized complexity conjecture. The following statements hold:

Stein-degree boundedness conjecture over non-closed fields. The quantity

Stein-degree boundedness conjecture. The quantity $\operatorname{sdeg}(S/Z)$ is bounded from above depending only on $d$ and $t$.

Let $d$ and $I$ be positive integers, and let $(X,B)$ be a projective $d$-dimensional klt log Calabi–Yau pair such that $I(K_X+B)\sim 0$. The birational automorphism group…