75 problems

Frobenius structure conjecture. The coefficient of $\vartheta_0$ in the product

Let $(Y,D)$ be a smooth log Calabi–Yau orbifold with maximal boundary, meaning that $D$ contains a $0$-stratum and $D\in|-K_Y|$. Let $\operatorname{QH}_{\log}^0(Y^{\dagger})$ denot…

Gromov–Witten integrality conjecture. The element

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…

Let $(X,D)$ be a log Calabi–Yau pair, let $(B,\mathsf{P})$ be its tropicalization, and let $A(X,D)$ be the topologically free $\mathbf{k}\llbracket NE(X)\rrbracket$-module with bas…

Consider a regular proper morphism defining a log Calabi–Yau fibration, and let a horizontal component of its boundary be a component that dominates the base. Its Stein degree is t…

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 $f$…

Mirror symmetric Gamma conjecture. The following identity should hold:

Connected Gromov–Witten integrality conjecture. The quantity

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 $Q$ be a connected component of the coarse moduli space of triples $(X,E,\Theta)$, where $X$ is a connected smooth projective complex variety, $E\in|-K_X|$ is an snc divisor wi…

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…

Hodge number duality conjecture. One should have

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…

Ducat's conjecture. The pair satisfies

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…