348 problems

Mirror symmetry conjecture. On the A-side, there are three non-birational Calabi–Yau threefolds with isomorphic derived categories.

Gopakumar–Vafa integrality conjecture. The BPS invariants $n_{h,\beta}(X)$ defined by this equation should be integers for all $h\geq 0$ and $\beta\neq 0$.

Let the Calabi–Yau threefolds under consideration be those represented within the Kreuzer–Skarke database. Reid's fantasy. All Calabi–Yau threefolds in this database are birational…

Let $X\subset\mathbb{P}^4$ be a generic quintic threefold. A rational curve means a curve on $X$ birational to $\mathbb{P}^1$, and its degree is measured with respect to the embedd…

Let $X\rightarrow X_0\leadsto Z$ be a conifold transition, and let $\check{X}$ and $\check{Z}$ be mirror Calabi-Yau threefolds to $X$ and $Z$, respectively. Morrison's conjecture.…

Semi-ampleness conjecture. Any nef integral divisor on a SCY3 is semi-ample.

Integral homology basis conjecture. The set $\{v_0,v_1',v_2,v_3\}$ is the image under $\iota_{\omega}$ of a $\mathbb Z$-basis of $H_3(X_{\phi}(\mathbb C),\mathbb Z)$.

Let $X$ be a polarized Calabi–Yau $3$-fold, let $\overline{\Omega}_\gamma^X\in\mathbb{Q}$ be its Donaldson–Thomas invariant, and define the BPS invariant…

Let $Y$ be a Calabi–Yau threefold with a fibration structure. A Tyurin degeneration is a degeneration of another Calabi–Yau threefold into normal crossing (quasi-)Fano components,…

Let $g_{50}$ be the modular form whose associated periods are well-defined modulo contributions from its residues. These boundary terms are considered modulo magnetic modular forms…

Let $X_{0}$ be a Calabi–Yau threefold over an algebraically closed field $k$ of characteristic $p$. Write $h^{1,0}$ for its Hodge number and $h_{W}^{1,2}$ for the corresponding cry…

Modularity conjecture. Any rigid Calabi–Yau threefold $X$ defined over $\mathbb{Q}$ is modular: its $L$-series $L(X,s)$ coincides up to finitely many Euler factors with the Mellin…

Let $X$ be the Calabi–Yau threefold in the paper, and let $\mathcal{M}$ be the moduli stack of objects $E\in D^b\operatorname{Coh}(X)$ satisfying … For $[E]\in\mathcal{M}$, let $G$…

Let $X$ be a Calabi–Yau threefold with $pi_1(X)=0$. For $c\in H^4(X,\mathbf{Z})$, call $c$ a DRY class if there exist an ample class $H\in H^2(X,\mathbf{R})$ and an integer…

Wilson's conjecture. The Kähler cone of $X$ should be rational and finitely generated in $N^1(X)$.

Let $X$ be a smooth Calabi–Yau threefold arising from the Kreuzer–Skarke database, and let a non-trivial Hilbert basis element be a divisor class in the Hilbert basis under conside…

Let $X$ be a smooth projective threefold that is the complete intersection of a quadratic and a quartic hypersurface in $\mathbb{P}^5$, and let $H$ be an ample class. For…

Let $X=Gr(2,7)[1^{7}]$ be the Calabi–Yau manifold obtained from the Grassmannian $Gr(2,7)$, and let $Y$ be the Pfaffian variety in $\mathbb{CP}^{20}\cap\mathbb{CP}^{6}$. Rødland's…

Let $S$ be a K3 surface with symplectic involution $g$, let $X=(S\times E)/\mathbb{Z}_2$, and let $\beta=(\gamma,d)\in H_2(X,\mathbb{Z})$ be a curve class. Define…

Katz's genus-zero GV/DT conjecture.

Let $O_5$ be the double cover of $P^3$ branched over the octic displayed in the source. For each prime $p>2$, let $a_p$ denote the eigenvalue for $T_p$ of the newform of weight $4$…

Douglas–Reinbacher–Yau conjecture. There exists a rank-$r$ stable reflexive sheaf $\mathcal E$ on $X$ with respect to some polarization such that $c_1(\mathcal E)=0$,…

A conifold transition of a smooth Calabi–Yau 3-fold $X$ is a flat degeneration to a nodal Calabi–Yau 3-fold followed by a small resolution. If the mirror Calabi–Yau 3-folds $Y^*$ a…

MNOP conjecture. The rank-one Donaldson–Thomas theory of $X$ should be equivalent to the Gromov–Witten theory of $X$.

Joyce–Song multiple cover conjecture. We have