61 problems

Pandharipande's Gopakumar--Vafa integrality conjecture. The primary Gromov--Witten theory of threefolds is governed by certain integers, called Gopakumar--Vafa invariants or BPS in…

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…

Katz's genus-zero GV/DT conjecture.

Let $X$ be a polarized Calabi-Yau threefold, let $\beta$ be a curve class, and let $S(X,\beta)$ be the moduli space whose Donaldson–Thomas invariant is … Let $N_\beta(X)$ denote th…

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 $d$ be a positive integer, and consider the appropriate moduli dg-stack of derived objects on ${\bf P}^2$ with characteristic numbers … The objects may instead need to be regar…

Let $n_{\mathbf d}^g$ denote the Gopakumar–Vafa invariant associated with genus-$g$ curves of degree $\mathbf d$ in a Calabi–Yau threefold. Gopakumar–Vafa integrality conjecture. T…

Let $\mathbb{P}$ be a smooth projective surface, let $\widehat{\mathbb{P}}$ be the blow-up of $\mathbb{P}$ along $L$ with exceptional curve $C$, and let…

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…

Square-dependence conjecture. If $2\nmid\beta$, then $\mathsf{GV}_{\beta}$ should depend only on the square $\beta^2$.

Refined Gopakumar–Vafa conjecture. One has

Maulik–Toda conjecture. One has

The non-commutative resolution is associated with a singular compact Calabi–Yau threefold whose non-Kähler small resolutions contain curves representing torsion homology classes. K…

Cohomological GV/PT correspondence for local $\mathbb{P}^2$. The conjectural series

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

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

Let $X$ be a Calabi–Yau threefold, let $\chi(X)$ be its Euler characteristic, let $\gamma^0$ be the distinguished charge-lattice element, and let…

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…

Assume the hypotheses of Maulik and Toda hold so that higher-genus Gopakumar–Vafa invariants $n^g_\gamma$ can be defined from the moduli of sheaves. Let the corresponding local hig…

Assume the hypotheses of Maulik and Toda hold in this setting, so that higher-genus Gopakumar–Vafa invariants $n^g_\beta$ can be defined from the Hilbert–Chow morphism and perverse…

Symmetry conjecture. In general, $\mathsf{MT}_{\beta}(y,w)$ is a Laurent invariant polynomial in $w$, invariant under $w\leftrightarrow w^{-1}$. The analogous invariance in $y$ fol…

Let $X$ be a Calabi–Yau threefold equipped with an involution $\imath$ preserving the holomorphic volume form. For a curve class $\beta\in H_2(X)$, let…

Let $X$ be a holomorphic symplectic 4-fold with an ample divisor $omega$. Fix $n\in\mathbb{Z}$, $\beta\in H_2(X,\mathbb{Z})$, and insertions $\gamma_i\in H^{m_i}(X,\mathbb{Z})$. Fo…

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 $n_{0,\beta}(\gamma_1,\ldots,\gamma_n)$,…