36 problems

Let $X$ be a projective complex $3$-manifold, let $beta eq 0$ be an algebraic curve class, and let $tau{ki}(etai)$ denote descendent insertions with $etaito H^{li}(X,mathbb{Q})$. W…

Let $S$ be a K3 surface, let $beta eq 0$ be a curve class, and let $n{g,beta}(S)$ denote the corresponding BPS invariant. The class $beta$ has square $beta^2$ and a divisibility in…

Let $S$ be a smooth projective surface and let $L$ be a sufficiently ample line bundle on $S$. The Severi degree $N^{(S,L),delta}$ counts $delta$-nodal curves in $|L|$ through…

Let $X$) be a Calabi–Yau $3$-fold, let $\beta$ be a nonzero effective curve class, and let $\mathop{\rm DT}\nolimits'_{\beta}(X)$ denote the coefficient of $t^\beta$ in the reduced…

Let $\mu^1$ and $\mu^2$ be partitions, let $Q$ and $q$ be formal variables, and define … Write … Here $\mathcal W_{\mu^1}$ and $\mathcal W_{\mu^2}$ denote the corresponding one-leg…

The paper considers chambered invariants $n_{1,2}$ and $n_{2,1}$ obtained by counting solutions to the $(1,2)$ and $(2,1)$ ADHM vortex equations for real Cauchy–Riemann operators.…

The perverse Hodge number generating-series conjecture. The perverse Hodge numbers depend on $\beta$ only through $\beta^2$, and, writing ${^ph}^{i,j}(M_{\beta})={^ph}^{i,j}_d$, th…

Let a geometric local curve element and an equivariant local curve element be local curve elements of the same topological type in the framework of the paper. Geometric–equivariant…

Complex–almost complex comparison conjecture. This map is an isomorphism. The conjecture is intended to permit almost complex methods to study invariants of complex threefolds, but…

Complex product expansion conjecture. The element $(X,t^{[\cdot]})\in H^0_c(\mathcal Z^{\mathrm{semi\textrm{-}Fano}}(\operatorname{Cpx}_3))[[t^{H_2(X)}]]$ is an infinite product

Klemm–Mariño's formula. For all $g$ and $\beta$,

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…

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…

Smaller-ring refinement. The preceding proposition still holds with this smaller ring $R$ in place of the ring allowing all factors $(1-Q^j)^{-1}$; in particular,

Let $Y$ be a Calabi–Yau threefold, and consider the moduli space of stable pairs on $Y$, together with the moduli space of stable maps and the relevant Hilbert schemes. Maulik–Nekr…

Let $X$ be a Calabi–Yau threefold, and let $F$ be a stable coherent sheaf on $X$ of pure dimension $1$, with Chern character $\operatorname{ch}(F)=\beta+\chi$, where $\beta$ is the…

Let $X$ be the smooth projective threefold under consideration, let $H$ be an ample divisor, and let $b_2(X)$ denote its second Betti number. For fixed sufficiently large $n$, let…

Let $X$ be the smooth projective threefold under consideration, let $H$ be its ample divisor, and let $\beta$ and $m$ be the numerical data defining $v_n$. Set … Let the strong Bog…

Let $X=\operatorname{Tot}_{\mathbb P^1}(\mathcal O\oplus\mathcal O(-1)\oplus\mathcal O(-1))$ and let $P_{n,d}(X;T)$ be the equivariant stable-pair invariant, where $d$ is the degre…

Let $(S,E)$ be a log Calabi-Yau surface with respect to a class $\beta\in\operatorname{H}_2(S,\mathbb{Z})$. The total log BPS number $m^{tot}_\beta$ is defined by the multiple-cove…

Let $(A,H)$ be a principally polarized abelian threefold with Picard rank $\rho(A)=1$. For nonzero $(\beta,n)\in\mathbb{Z}^2$, define…

Curve-counting conjecture for rationality of $X_{5,6}$. For a generic point $p\in D_{1,0,\zeta,0,-1}$, one has $\#(p)=1$. Together with rationality of $X_{4,6}$, the source states…

Curve-counting conjecture for $X_{4,6}$. For a generic point $p\in E^4$, one has $\#(p)=1$. Moreover, for a generic point $p\in D_{0,1,2,1}$, one has $\#(p)=1$. This conjecture imp…

Let $X$ be as in the definition of local Gopakumar–Vafa invariants, let $\beta$ be an effective one-cycle class, and let $\gamma\in\operatorname{Chow}_{\beta}(X)$. For…

Let $X$ be a Calabi–Yau threefold, let $\beta$ be an effective one-cycle class, and let $gamma in operatorname{Chow}{beta}(X)$ be an irreducible one-cycle. Denote by…