33 problems

The JMGS conjecture. The small $J$-function satisfies

Let $X$ be a Calabi–Yau threefold with only terminal conifold singularities and no Kähler small resolution. Let $\mathcal{S}$ be the set of all small resolutions $\widehat{X}$ of…

Let $X$ be a Calabi–Yau threefold, fix $g\ge0$, and suppose $I_{1-g}(X,\beta)$ parametrizes ideals of local complete intersection curves of arithmetic genus $g$. Let…

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 smooth projective Calabi–Yau $3$-fold. For a curve class $\beta \in H_2(X, \mathbb{Z})$ and a non-negative integer $g \in \mathbb{Z}_{\geq 0}$, let…

Refined Gopakumar–Vafa conjecture. One has

Maulik–Toda conjecture. One has

Refined Gopakumar–Vafa/Pandharipande–Thomas correspondence. One should have

The Gopakumar–Vafa/Pandharipande–Thomas correspondence. One should have

Virtual Clemens conjecture. For the purpose of computing genus zero enumerative invariants on Calabi–Yau threefolds $X$, one may assume that there are only finitely many isolated r…

Let $X$ be a holomorphic symplectic $4$-fold and let $M_\beta$ be the moduli scheme of one-dimensional stable sheaves $F$ on $X$ with $\operatorname{ch}_3(F)=\beta$ and $\chi(F)=1$…

Let $X$ be a holomorphic symplectic $4$-fold, let $\beta\beta\in H_2(X,\mathbb{Z})$ be a primitive curve class, and let $n_{0,\beta}(\gamma_1,\ldots,\gamma_n)$,…

Let $X$ be the smooth projective variety under consideration, let $q$ and $t$ be formal variables, let $t^\beta$ record effective curve classes $\beta$, and write…

Let $X$ be the smooth projective variety under consideration, let $N^{\operatorname{GW}}_{g,\beta}(X)$ denote its Gromov–Witten invariants, let $u$ be a formal variable, and let…

Let $(X,\omega)$ be a closed symplectic $6$-manifold. For every $A\in H_2(X,\mathbf Z)$ with $A\neq 0$ and $c_1(A)=0$, let $\mathrm{BPS}_{A,g}(X,\omega)$ be the numbers defined by…

The quintic quantum K-theory conjecture. The small $J$-function of the quintic is expressed linearly in terms of the GV-invariants by

Let $X$ be a smooth projective Calabi–Yau $4$-fold, let $\beta\in H_2(X,\mathbb{Z})$, and let $M_1(X,\beta)$ be the moduli space of one-dimensional stable sheaves $F$ on $X$ with…

Let $X$ be a Calabi–Yau $4$-fold and let $M_1(X,\beta)$ be the moduli scheme of one-dimensional stable sheaves $F$ with $[F]=\beta\in H_2(X,\mathbb Z)$ and $\chi(F)=1$. Let…

Let $(X,\omega)$ be a smooth projective Calabi–Yau $4$-fold, let $P_{n,\beta}(\gamma)$ be its stable-pair invariants with $\gamma\in H^4(X,\mathbb Z)$, and let…

Let $C$ be a flopping curve, let $r$ be a positive integer, and let $\operatorname{GV}_{C,r,0}$ be the associated monodromic mixed Hodge structure. Write $n_{C,r}(q^{1/2})$ for its…

Stable-pair genus-one correspondence conjecture. For a suitable choice of orientation,

Klemm–Pandharipande's genus-zero integrality conjecture. The invariants $n_{0,\beta}(\gamma_1,\ldots,\gamma_n)$ are integers. This is the genus-zero integrality prediction for Cala…

Katz's genus-zero GV/DT conjecture.

Let $X$ be a smooth projective Calabi–Yau threefold, let $v=(\beta,m)$ with $\beta eq 0$, let $sigmain U(X)$, and let $gammainoperatorname{Chow}X(\beta)$. Suppose that the moduli sta…

Let $X$ be a Calabi–Yau threefold, let $\beta>0$, and let $\operatorname{Sh}_{\beta}(X)$ be the moduli space used to define the Gopakumar–Vafa invariants, equipped with its $d$-cri…