55 problems

Let $(X,D)$ be a log Calabi–Yau surface pair with $D=D_1+\cdots+D_l$, and define … where $\mu$ is the Möbius function. Klemm–Pandharipande integrality conjecture. When $l=2$, the q…

Let $Y$ be a Calabi–Yau $4$-fold, let $\beta\in H_2(Y,\mathbb{Z})$ be a curve class, and let $\mathbf{M}_{\beta}(Y)$ be the moduli space of one-dimensional stable sheaves $F$ on…

Let $X$) be a projective Calabi–Yau $4$-fold and let $L$ be a line bundle on $X$. Define … Here $M(q)=\prod_{i=1}^{\infty}(1-q^i)^{-i}$ is the MacMahon function. Cao–Kool's conject…

A five-dimensional reflexive polytope is a lattice polytope whose only interior lattice point is the origin and whose polar dual is also a lattice polytope. Such a polytope gives r…

Let $t_1,t_2,t_3,t_4,y,q$ be formal variables satisfying $t_1t_2t_3t_4=1$. For any formal variable $x$, write $[x]=x^{\frac12}-x^{-\frac12}$, and let $\operatorname{Exp}$ denote th…

Genus-zero GW/DT4 conjecture. For a suitable choice of orientation,

Let $X$ be a nonsingular, projective Calabi–Yau $4$-fold and let $n_{1,\beta}$ be the genus-one invariants defined from the genus-one Gromov–Witten invariants $N_{1,\beta}$, togeth…

Genus-zero integrality conjecture. The invariants $n_{0,\beta}(\gamma_1,\ldots,\gamma_n)$ are integers. This is the proposed Calabi–Yau $4$-fold analogue of the Aspinwall–Morrison…

Let $S$ be a toric surface, let $\bar X=\mathbb P(\mathcal O_S\oplus K_S)$, and let $K_{\bar X}$ be its canonical bundle. Let $J_i,J_j,J_k,J_l$ be divisor classes on $\bar X$, let…

Let $X$ be a toric Calabi–Yau $4$-fold, let $\mathscr L$ be a $\mathbb T$-equivariant line bundle, and let $\mathsf{DT}_{X,\mathscr L}(m,q)$ denote the generating function of zero-…

Let $X$ be a projective Calabi–Yau $4$-fold. Write $\operatorname{Hilb}^n(X)$ for the Hilbert scheme of $n$ points on $X$, let $L\to X$ be a line bundle, and distinguish Gromov–Wit…

Let $E\rightarrow\mathcal{M}$ be a variation of polarized Hodge structure of weight $D=2k$. For a positive integer $L$, define … Here $\mathcal{M}$ is the complex-structure moduli…

Let $X$ be a smooth Calabi–Yau fourfold, and let the vertical subgroup be the subgroup of the middle cohomology generated by products of divisor classes. Its intersection pairing c…

Let $G=\mathbb{Z}_2\times\mathbb{Z}_2<\mathrm{SO}(3)<\mathrm{SU}(4)$ and set $q_{(2,2)}:=q_{00}q_{10}q_{01}q_{11}$. Define $\mathcal{F}_{2,2}$ and…

The orbifold vertex conjecture. There exist orientations such that

Let $G<\mathrm{SU}(4)$ be a finite subgroup whose elements have age at most $1$, and let $X\to\mathbb{C}^4/G$ be the crepant resolution given by the Nakamura $G$-Hilbert scheme. Wr…

Let $X$ be a Calabi-Yau $4$-fold and $S\subseteq X$ a local complete intersection surface. For an extension of local Artinian schemes…

Let $f:\mathcal{X}\to\mathcal{B}$ be a smooth projective morphism with connected fibres of relative dimension $4$ to a smooth connected affine scheme $\mathcal{B}$, with…

Cao–Toda's genus-one descendent correspondence conjecture. Via some suitable choice of orientation on the moduli space, one has, for all $\gamma\in H^2(X,\mathbb Z)$,

Cao–Maulik–Toda's genus-zero correspondence conjecture. Via some suitable choice of orientation on the moduli space, one has

Let $X$ be a Calabi–Yau fourfold, let $\mathcal{A}=\operatorname{Coh}(X)$, and let $\tau$ be a Gieseker stability condition. Write $\mathcal{M}^{\operatorname{ss}}_\alpha(\tau)$ an…

Wall-crossing conjecture. If $\Theta\in L_-^-(k)$ or $L_+^-(k)$ with $k\geqslant1$, then there exist choices of signs such that

Cao–Kool–Monavari's conjecture. There exist choices of signs such that

Let $(X,\omega)$ be a polarized Calabi–Yau 4-fold with a projective surjective morphism $\pi:X\to B$ to a variety $B$. Let $i:D\hookrightarrow X$ be a smooth divisor of the form…

Let $X$ be a Calabi–Yau 4-fold and let $\beta$ range over effective curve classes. For $\theta\text{?}$ [the source does not define the notation $\mathrm{PT}(X)(\gamma)$, $q$, or…