12 problems

Joyce–Song multiple cover conjecture. We have

Let $X$ be the variety under consideration, let $N_{n,\beta}$ denote the stable-pair coefficients in the logarithm of the stable-pair partition function, and let $n_0^\gamma$ denot…

Multiple Cover Formula.

Let $S$ be a K3 surface, let $g\geq 0$, let $m$ be a positive integer, let $\alpha$ be a tautological class, and let $\gamma_1,\ldots,\gamma_n\in H^*(S,\mathbb{Q})$ be homogeneous…

Let $S$ be a K3 surface, let $\beta$ be a primitive curve class, let $m$ be a positive integer, and let $\gamma_1,\ldots,\gamma_n$ be cohomology insertions. Let…

Jiang's twisted multiple-cover conjecture. If $H^{0,1}(S)=H^{0,2}(S)=0$, there exist rational numbers $\operatorname{VW}^{\operatorname{tw}}_{\alpha_i}({\mathfrak S})$ such that

Let $Y$ be a compact Calabi–Yau 3-fold and let $\beta\in H_2(Y,\mathbb Z)$. Denote by $\mathrm{GW}_{0,\beta}(Y)$ the genus-zero Gromov–Witten invariant and by…

Let $Y$ be a smooth projective Calabi–Yau $3$-fold. For a curve class $\beta$ and integer $n$, let $N_{n,\beta}\in\mathbb Q$ be the generalized Donaldson–Thomas invariant counting…

Let $A$ be an abelian threefold, and let $\beta\in H_2(A,\mathbb Z)$ be a curve class of type $(d_1,d_2,d_3)$. Write $\mathsf{DT}^{\mathrm{red}}_{n,(d_1,d_2,d_3)}$ for the reduced…

Let $X=S\times E$ be the product of a nonsingular projective K3 surface $S$ and an elliptic curve $E$, with $E$ acting by translation on the second factor. For…

Let $N_{n,\beta}$ denote the invariants appearing in the Gopakumar–Vafa-type product expansion of the stable-pair generating series, and let $(n,\beta)$ denote the common divisibil…

Toda's K3 multiple cover conjecture. We have