19 problems

Let $X\subset\mathbb{P}^4$ be a smooth quintic 3-fold, and assume the strong Bogomolov–Gieseker conjecture used to construct the heart $\mathcal{A}_G$ and central charge…

Let $\pi\colon\mathcal W\to\mathbb P^{1}$ be a family of quintic mirror threefolds. Let $N_g(d)$ be the genus-$g$ Gromov–Witten invariant of degree $d$ of a general quintic threefo…

Consider the extended GLSM in the positive phase $\zeta>0$, with GLSM $I$-function $I^+_w(q,z)$ and GLSM small $J$-function $J^+_w(Q,z)$. Let $I^{\mathrm{CY}}_0(q,z)$ and…

Let $F^{\mathrm{LG}}_{0,1}(\tau)$ be the generating function of genus-zero open FJRW invariants of $(W_5,\mu_5)$, and let $\mathcal{T}^{\mathrm{LG}}(t)$ and $I^{\mathrm{LG}}_k(t)$…

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…

The quintic quantum K-theory conjecture. The small $J$-function of the quintic is expressed linearly in terms of the GV-invariants by

Let $G_{g,n}^{\mathrm{BCOV}}$ be the set of genus-$g$, $n$-leg stable graphs with the BCOV edge types and solid or dotted legs. Define $f_{g,n}^{\mathrm{BCOV}}$ as the sum over the…

Let $G_g^{\mathrm{BCOV}}$ be the set of genus-$g$ stable graphs for the quintic threefold, with solid, half-dotted-half-solid, and dotted edges. Place the propagators…

Yamaguchi–Yau genus-two mirror conjecture. The function $F^{GW}_2(Q)$ is given by

Let $S_{g,d}$ and $S'_{g,d}$ be the indexing sets used in the paper, let $B_{g,d}(\rho)$ and $C_\rho$ be the corresponding terms and coefficients in the degeneration equations, and…

Let $Q$ be the quintic threefold in $\boldsymbol{^4}$. Let $N_{g,d}$ denote its Gromov–Witten invariants, and let the degeneration formula for $(V,W)=(\mathbb{P}^4,Q)$ relate the r…

Low-degree GW algorithm conjecture. Applying the localization vanishing to this data determines $N_{g,d}$ provided that

GW effective-algorithm conjecture. By choosing $d_0$ and $d_\infty$ positive, these relations provide an effective algorithm to determine all genus GW invariants $N_{g,d}$, provide…

Let $\Theta_{g,k}$ denote the FJRW invariants for the quintic indexed by genus $g$ and $k$, and let the relations … . Effective determination conjecture. The equations … , effectiv…

Let $X$ be a general quintic threefold, and let $d$ be the degree of a smooth rational curve on $X$. The scheme parametrizing smooth rational curves of degree $d$ on $X$ records th…

Let $\overline{M}$ be the quintic three-fold, let $D\subset\overline{M}$ be the smooth hyperplane section used in the paper, and set $M=\overline{M}\backslash D$. Write…

Let $Q\subset\mathbb{P}^4$ be a smooth quintic, and let $N_{g,d}(Q)$ denote its genus $g$, degree $d$ Gromov–Witten invariant. Let $N'_{h,d}(Q)$ be the reduced invariant defined by…

Strong Bogomolov–Gieseker conjecture. One has

Let $X$ be a very general quintic threefold in $\mathbb{P}^4$, and let $\phi\colon X\dashrightarrow X$ be a rational self-map. Self-rational-map conjecture. The map $\phi$ cannot h…