19 problems

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 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 $X$ 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 $\beta$ be a curve class, let $(S,E)$ be general with respect to $\beta$, and let $P,P'\in E(\beta)$ be $\beta$-primitive points. Primitive-point independence conjecture. … Thi…

Let $(S,E)$ be a rational log Calabi-Yau surface with smooth divisor and let $\beta\in\operatorname{H}_2(S,\mathbb{Z})$. For $P\in E(\beta)$, let $m^P_\beta$ denote the log BPS num…

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…

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…

Let $N_{n,\beta}$ be the curve-counting invariants introduced through the PT product expansion, let $n\in\mathbb Z_{>0}$, and let $\beta$ be an effective curve class. Write…

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…