30 problems

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. 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…

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 $f:\mathcal{X}\to\mathcal{B}$ 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 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…

Klemm–Pandharipande integrality conjecture.

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

Let $X$ be a Calabi–Yau $4$-fold and let $\mathcal A=\operatorname{Coh}(X)$, with the moduli, stability, orientation, homology, and virtual-class data specified in the universal en…

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…

In the setting of the wall-crossing conjecture, let $P^{\mathrm{JS}}_{n,\beta}(\gamma)$ denote the Joyce–Song chamber stable-pair invariant, and let $n_{0,\beta}(\gamma)$ be the ge…

Let $(X,\omega)$ be a smooth projective Calabi–Yau $4$-fold, let $\beta\in H_2(X,\mathbb Z)$ and $n\in\mathbb Z_{\geqslant0}$, and choose a generic $t\in\mathbb R_{>0}$. Let…

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 $X=\operatorname{Tot}_{\mathbb P^1}(\mathcal O(-1)\oplus\mathcal O(-1)\oplus\mathcal O)$, and let $\mathcal Z_X(y,q,Q)=\sum_{n,d}P_{n,d[\mathbb P^1]}(\mathcal O,y)q^nQ^d$ be it…

Let $\lambda,\mu,\nu,\rho$ be finite plane partitions, with at most two of them non-empty in the stable-pairs case. Let…

Let $X$ be a smooth projective Calabi–Yau fourfold and let $\beta\in H_2(X,\mathbb Z)$. For a line bundle $L$ on $X$, write $L^{[n]}$ for its tautological bundle on the relevant mo…

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…

Let $X$ be a smooth projective Calabi–Yau 4-fold, let $I_n(X,\beta)$ be the Hilbert scheme of one-dimensional subschemes with Chern character $(0,0,0,\beta,n)$, and let…

Let $X$ be a smooth projective Calabi–Yau $4$-fold, let $\beta\in H_2(X,\mathbb{Z})$ be an irreducible curve class, and let $\gamma_1,\ldots,\gamma_l$ be integral cohomology classe…

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…

Let $X$ be a sextic $4$-fold, let $\beta$ be a curve class, and let $\rho\in H^{4}(X,\mathbb{Z})$. Denote by $\mathrm{GW}_{0,\beta}(\gamma)$ the genus-zero Gromov–Witten invariant…