34 problems

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…

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…

Let $Q\subset \mathbb{P}^4$ be the Calabi–Yau quintic threefold, and let $N_{g,d}$ denote its Gromov–Witten invariants. Let $(\mathbb{P}^4,Q)$ be the corresponding relative pair. T…

Let $F=\operatorname{\mathbb{Q}}(\sqrt{5})$, let $\widetilde{X}$ be the proper and smooth $\operatorname{\mathbb{Z}}[\frac{1}{30}]$-scheme described in the paper, and let … Suppose…

Tyurin's conjecture. All rank $2$ stable bundles $\mathcal E$ on a general quintic threefold are infinitesimally rigid:

Let $n_d$ denote the number of rational curves on the quintic threefold of degree $d$, and form their generating function. A hypergeometric series is related to this generating fun…

Kontsevich's conifold monodromy conjecture. For the monodromy around the conifold point, the kernel $\Psi$ is the complex

Let $X$ be the quintic hypersurface in ${\mathbb P}^4$, with $P_1$ denoting the conifold point in the moduli space of complexified Kähler forms. Let ${\Scr K}^K_{\Scr E}$ denote th…

Let $I=(1,1,1,1,1)$, and let … be the Fermat quintic. A point $(X_{0},X_{1},X_{2},X_{3},X_{4})\in\mathcal{V}_{I}(\mathbb{Z})$ is non-trivial if the greatest common divisor of its c…

Consider the maximal point insertion theory for the open Fermat quintic with minimal symmetry group, with boundary twists $(1,1,1,1,1)$ and $(3,3,3,3,3)$. Its boundary-only interse…

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 $I^{\mathrm{LG}}_k(t)$…

Candelas' rational-curve conjecture. The number $n_j$ is the number of rational curves of degree $j$ on a general quintic threefold in ${\bf P}_4$. The claim connects mirror-symmet…

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…

Candelas et al.'s conjecture. The numbers $a_n$ are integers equal to the virtual number of degree-$n$ rational curves on a generic quintic threefold in $\mathbb{P}^4$.

Let $X$ be the quintic threefold, let $\Phi_\alpha=x^\alpha$ for $\alpha=0,1,2,3$ be the chosen basis of $K(X)$, and let $I(t)$ be the quintic quantum K-theory I-function. Write th…

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

Maulik–Pandharipande's algorithm conjecture. The second algorithm determines $N_{g,d}$ for all genera $g$ and degrees $d$.

Let $\mathcal F_g(q,\bar q)$ be the non-holomorphic completion of the Gromov–Witten potential $F_g(q)$, and let $F_g^{\mathrm{orb}}(\tau^{\mathrm{orb}})$ be the potential of the co…

Yamaguchi–Yau's holomorphic anomaly equations. The potentials satisfy

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…

Let $H_{\mathbf B}:=\operatorname{span}\{\bar\psi,\tilde\phi\}$ be the state space, and let $G_g$ be the set of genus-$g$ stable graphs. For each $\Gamma\in G_g$, use the BCOV edge…

Complete intersection surface conjecture. As soon as $X$ contains a complete intersection surface of bidegree $(a,b)$ with $a,b<5$, it also contains a $2$-plane. Consequently, the…

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