13 problems

Let $X$ be a smooth projective variety, let $V$ be a fixed sheaf on $X$, and let $P$ be a moduli space of pairs consisting of a sheaf $F$ and a map $V\to F$, with universal pair…

Let $X$ be a smooth projective variety and let $M$ be a moduli space of sheaves on $X$ with virtual fundamental class $[M]^{\mathrm{vir}}$. Let $\mathbb{D}^X$ be the descendent alg…

Let $X$ be a simply-connected smooth projective 3-fold, let $\gamma\in H^{p,0}(X)$ with $p=2$ or $p=3$, and let $D\in\mathbb D_{\textup{PT}}^{X}$. For $k\geq 0$, the stable-pairs d…

DT/PT correspondence conjecture. For any such data,

Oblomkov–Okounkov–Pandharipande conjecture. If all insertions are stationary, meaning $\alpha_i\in H^{>0}(X)$, then the renormalized invariant is a rational function of $q$ regular…

Equivariant DT descendent algebra conjecture. For $\gamma_i\in H^*_{\mathsf{T}}(X,\mathbb{C})$ and $\beta\in H_2(X,\mathbb{Z})$,

GW descendent polynomiality conjecture. The series

Stationary GW/PT correspondence conjecture.

Let $X$ be a nonsingular projective $3$-fold and let $\beta\in H_2(X,\mathbb Z)$ be a nonzero effective class. Let $d=\operatorname{div}(\beta)$ be the divisibility of the image of…

Relative descendent GW/Pairs correspondence. Under the variable change $-q=e^{iu}$,

Descendent GW/Pairs correspondence. Under the variable change $-q=e^{iu}$,

Relative descendent GW/Pairs correspondence.

Stable-pairs rationality conjecture. The partition function $\mathsf Z_{\beta}^X\left(\prod_{j=1}^k\tau_{i_j}(\gamma_j)\right)$ is the Laurent expansion of a rational function in…