136 problems

Stable-pair rationality and symmetry conjecture. The series $Z_\beta^\mathsf{PT}(q\mid D)$ is the Laurent expansion of a rational function, more precisely of $q^{d_\beta/2}f(q)$ fo…

Virasoro constraints conjecture. For all $k\geq -1$ and $D\in\mathbb D_{\mathrm{PT}}^X$,

Let $X$ be the CHL geometry in the source, let $C'(\lambda)\subset X$ be the subscheme associated with a partition $\lambda$, and let $\operatorname{Hilb}^m_{C(\lambda)}$ be the co…

Let $(Y\mid\partial Y)$ be a logarithmic pair, with logarithmic Donaldson–Thomas and stable-pairs generating series, and let the degree-$0$ evaluation denote the degree-$0$ contrib…

Equivariant rationality conjecture. The series $\widetilde Z^{S}_{\beta}(q\mid s_1,s_2)$ is the Laurent expansion of a rational function in $q,s_1,s_2$. This is presented as an equ…

Let $X$ be a nonsingular projective 3-fold with only $(p,p)$-cohomology, let $\beta\in H_2(X,\mathbb{Z})$, and let $\mathbb{D}^X_{\mathrm{PT}}$ be the stable pairs descendent algeb…

Let $S$ be a nonsingular projective $K3$ surface, let $E$ be an elliptic curve, and set $X=S\times E$. For a positive class $\beta\in\operatorname{Pic}(S)$ and an integer $d$, let…

Boundedness conjecture. For every positive rational number $C$ there exist a positive integer $N>0$ such that, for every stable $n$-dimensional stable pair $(X,B)$ with $(K+B)^n=C$…

Pandharipande--Thomas correspondence conjecture. The stable-pair generating series matches the normalized Donaldson--Thomas generating series; in particular, it is rational and sat…

Nekrasov–Okounkov conjecture. The generating series of these K-theoretic stable-pair invariants coincides with the Laurent expansion of the M2-brane index of $Z$.

Let $(Y\mid\partial Y)$ be a toric threefold pair, let $\beta$ be a curve class, let $\bm\mu$ be a vector of partitions, and let $[[![\sigma,\vartheta,\upalpha][!]]$ be a stratum c…

Let $(Y\mid\partial Y)$ be a toric threefold pair, let $\beta$ be a curve class, let $\bm\mu$ be a vector of partitions, and let $\upalpha$ be an insertion. Write $d_\beta$ for the…

Let $M$ be a smooth projective threefold, $D\subset M$ a connected divisor, and let $\beta$ be a curve class. Let $A=\prod_{i=1}^r\tau_{\alpha_i-1}(\gamma_i)$ be a product of desce…

Let $M$ be the smooth projective threefold under consideration, let $\beta$ be a curve class, and let $A=\prod_{i=1}^r\tau_{\alpha_i-1}(\gamma_i)$ be a product of descendent insert…

Stable-pair/Donaldson–Thomas correspondence conjecture.

Let $\bold X$ be a logarithmic Calabi–Yau 3-fold, let $D^{[\beta]}$ be the product of the relevant Hilbert schemes of points on the interiors of the boundary divisors, and let … be…

Degree-two rationality conjecture. The series $Z_{d=2}^{\mathbb P^2}(q)$ is the Laurent expansion of a rational function in $q$. The claim is supported only by interpolation and th…

Virtual Euler characteristic rationality conjecture. The series $Z^{S}_{\beta}(q)$ is the Laurent expansion of a rational function of $q$. Viewed as a rational function, it satisfi…

Let $S$ be a cubic surface with $27$ marked lines $D_1,\ldots,D_{27}$, and set … where $\varepsilon\in\mathbb{Q}$ and $0<\varepsilon\ll 1$. Hacking–Keel–Tevelev's conjecture. The c…

Rationality, symmetry and pole-structure conjecture. For any $X$ and $T$ as above: (1) every $\mathbf z$-coefficient of the disconnected and connected descendent invariants is the…

Motivic PT generating-series conjecture. The generating series of the fiber-class invariants is given by the displayed infinite-product and plethystic-exponential formula.

Brini–Schüler refined stable-pairs/Gromov–Witten correspondence. One has

Let $\boldsymbol\mu=\{\mu_a\}_{a=1}^{4}$ be plane partitions with no PT$_0$ moduli, and let $\mathsf V^{\mathrm{DT}}_{\mu_1\mu_2\mu_3\mu_4}(q)$ and…

Let $X$ be a Calabi–Yau 4-fold with an algebraic torus $T$ acting while preserving the Calabi–Yau volume form, and let $L$ be a $T$-equivariant line bundle on $X$. For…

Let $X$ be a quasi-projective Calabi–Yau 4-fold, and let $\curly P=\curly P_v^{(q)}(X)$ be the relevant PT$_q$ moduli space. Assume that $\curly P$ is projective, so that its virtu…