70 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…

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…

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…

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

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…

Fix plane partitions $\boldsymbol{\mu}=\{\mu_a\}_{a=1}^{4}$ corresponding to $(\mathbb C^*)^3$-fixed subschemes of $\mathbb C^3$ of dimension at most one, viewed as asymptotic cros…

Let $(X,D)$ be the logarithmic pair, let $\beta$ be a curve class, and set … Let $\alpha=(\alpha_1,\ldots,\alpha_k)$ be a partition, let $\zeta_1,\ldots,\zeta_k$ be cohomology clas…

Let $(X,D)$ be the logarithmic pair, let $\beta$ be a curve class, and set … Let $\zeta_1,\ldots,\zeta_k$ be cohomology insertions and let $\bm{\mu}(\delta)$ be relative conditions…

Refined Gopakumar–Vafa/Pandharipande–Thomas correspondence. One should have

Canonical oriented d-critical structure conjecture. There exists a canonical oriented d-critical locus structure on $P_n(X,\beta)$ inducing $\phi_{n,\beta}$, with…

The Gopakumar–Vafa/Pandharipande–Thomas correspondence. One should have

Let $G=\mathbb{Z}_r<\mathrm{SU}(2)<\mathrm{SU}(4)$ or $G=\mathbb{Z}_2\times\mathbb{Z}_2<\mathrm{SO}(3)<\mathrm{SU}(4)$. Let $X\to\mathbb{C}^4/G$ be the Nakamura $G$-Hilbert crepant…

Let $X_5$ be the quintic threefold, let $H$ be its hyperplane class, and set $\kappa=H^3=5$. Write $\overline{\Omega}_H$ for the rank-zero invariant with Chern-character arguments…

Let $X$ be a holomorphic symplectic 4-fold and let $\beta\in H_2(X,\mathbb{Z})$ be an irreducible curve class. Let $P_n(X,\beta)$ be the moduli space of stable pairs and…

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…

The marked relative GW/PT correspondence. For all $\gamma\in H^{\ast}((X,D)^r)$, $Z^{(X,D)}_{\mathsf{PT},\beta}(\lambda\mid\tau_{\alpha_1-1}\cdots\tau_{\alpha_r-1}(\gamma))$ is the…

The relative GW/PT correspondence. $Z^{(X,D)}_{\mathsf{PT},\beta}(\lambda\mid\tau_{\alpha_1-1}(\gamma_1)\cdots\tau_{\alpha_r-1}(\gamma_r))$ is the Fourier expansion of a rational f…

The GW/PT correspondence. $Z^{X}_{\mathsf{PT},\beta}(\boldsymbol{\tau}_{\alpha-1}(\boldsymbol{\gamma}))$ is the Fourier expansion of a rational function in $p$, and under $p=e^z$,

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 a nonsingular projective 3-fold, let $\beta\in H_2(X,\mathbb{Z})$, and let … be the stable pairs descendent series, where $\gamma_i\in H^*(X)$ and $k_i\geq 0$. Stable pa…

Let $(X,D)$ be a logarithmic pair, let $\beta$ be a curve class, let $\mu$ be a relative insertion, and let $\tau_{k_i}(\gamma_i)$ be descendent insertions. Denote the logarithmic…

Let $X=\mathcal O_{\mathbb P^1}(-1,-1,0)$ and let $T$ be the Calabi–Yau torus, with equivariant parameter $\lambda_3$. Let $P^{\mathrm{JS}}_{n,d}$ denote the $T$-equivariant Joyce–…