903 problems

Strominger–Yau–Zaslow conjecture. The mirror $Y^{\vee}$ is the moduli space of certain A-branes in $Y$, and the mirror functor in homological mirror symmetry is given by a fiberwis…

Let $X$ be a smooth algebraic variety whose bounded derived category of coherent sheaves is equipped with a Frobenius structure, and let $D^b\operatorname{Coh}(X)$ denote that boun…

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

Let $V$ be a cluster variety and let $V^\vee$ be its mirror cluster variety. Write $V^\vee(\mathbb{Z}^T)$ for the tropical points of $V^\vee$. Fock–Goncharov's cluster duality conj…

Let $M_{n,a}$ be a smooth hypersurface of degree $a\le n+1$ in $\mathbb{CP}^{n+1}$, let $D_{n,a,i}$ be a union of $i\le n+2$ generic hyperplanes, and let…

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.…

Let $X$ be a smooth algebraic curve of genus $g\ge 2$, let $L\in\operatorname{Pic}(X)$ have degree $1$, and let $\Gamma=\operatorname{Pic}^0(X)[n]$. Denote by … the moduli space of…

Let $\Delta$ and $\Delta^*$ be dual reflexive polytopes. Consider anticanonical hypersurfaces in the corresponding toric varieties, together with their crepant resolutions when nee…

Let $f_n$ be the chain-type invertible polynomial, let $S_n$ and $L_{f_n}$ be the associated ring and grading group, and let $\omega_1,\dots,\omega_n$ be positive rational numbers…

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 $H_3(X_{\phi}(\mathbb C),\mathbb Z)$.

Teleman's conjecture. A Hamiltonian $G$-action on $X$ should be mirror to a holomorphic fibration

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 $M_N^k$ be a degree-$k$ hypersurface in ${\mathbb P}^{N-1}$, let $w({\cal O}_{h^a}{\cal O}_{h^b}|\prod_{p=0}^{N-2}({\cal O}_{h^p})^{m_p})_{0,d}$ denote its genus-$0$ multi-poin…

Let $J$ be a compact $G_2$ manifold, and consider Type IIA and Type IIB string compactifications on $J$. At the level described, both compactifications have the same matter content…

Let $(\Delta,\{\Delta_i\}_{i=1}^{r})$ and $(\nabla,\{\nabla_i\}_{i=1}^{r})$ be dual nef-partitions, and let $X$ and $X^{\vee}$ be smooth maximal projective crepant partial desingul…

Consider curves with $y=\log z$ of the form … where $P_1$ and $P_2$ are coprime polynomials. Let $\psi^{\vee}(z;b)$ be the dual wavefunction, and write $\hat{x}^{\vee}=y\cdot$ and…

Let $X$ be a smooth Fano variety with a normal crossing anti-canonical complement $D=D_1+\ldots+D_r$, and let $U=X\setminus D$. Suppose that $U^\vee$ is a mirror dual open Calabi–Y…

Mirror-family conjecture. Based on numerical evidence, the mirror of the K3 family is given by a certain family of double covers over a del Pezzo surface of degree $6$.

Mori cone generation conjecture. The Mori cone $M$ of $X_\psi$ is generated by the classes of the curves $\ell_{\mathbf{t}}$, $\sigma_{\mathbf{s},\mathbf{t}}$, and…

Closed string mirror symmetry conjecture. There are equivalences, as Frobenius manifolds,

Hochschild cohomology quotient conjecture. There exists a $\mathbb{C}[s]$-module structure on $HH^*(\mathcal{W}(X))$ such that $HH^*(\mathcal{W}(\underline{X}))$ is isomorphic to t…

Q-Gorenstein–mutation conjecture. $\mathbb{Q}$-Gorenstein equivalence classes of projective Fano varieties with mild singularities correspond to mutation equivalence classes of rig…

The remodeling conjecture concerns a toric Calabi–Yau threefold and its mirror curve, on which the Eynard–Orantin topological recursion is defined. Remodeling conjecture. The open…

Let $X$ be a smooth Fano variety. A toric Landau–Ginzburg model existence conjecture asserts that $X$ has a Laurent polynomial satisfying the period, Calabi–Yau, and toric conditio…

Let $W$ be an invertible polynomial, let $W^T$ be its Berglund–Hübsch transpose, and let $G_W$ be the maximal group of diagonal symmetries of $W$. LG/Saito–Givental mirror equivale…