382 problems

The big quantum cohomology of G/P is generically semisimple.

For every integer $p\ge 2$, determine whether the higher-genus closed Gromov–Witten potentials of the toric Calabi–Yau threefold…

Let $h$ be the hyperplane class of $CP^2$, and let $w({\cal O}_{h^a}{\cal O}_{h^b}\mid(x^0,x^1,x^2))_0$ denote the generating function of multi-point virtual structure constants. D…

Let $\mathcal{M}$ be an $SU(2)$ Hitchin moduli space with prescribed singularities, and let $\hat{\mathcal{M}}$ be its SYZ mirror, an $PGL(2)$ Hitchin moduli space with prescribed…

Let $X$ and $Y$ be “mirror” Calabi-Yau manifolds. There should exist a base manifold $B$ and surjective continuous maps … with fibers $L_q=\pi^{-1}(q)$ and $F_q=f^{-1}(q)$, and a c…

Let $(Y,w)$ be a Landau–Ginzburg model of Fano type arising, together with a tame compactification, as a mirror of a projective Fano manifold $X$ of dimension $n=\dim Y$. Let…

Let the Higgs and Coulomb branches of a 3d $\mathcal{N}=4$ supersymmetric gauge theory be mirror dual symplectic singularities, and let $\mathcal{O}$ denote the category associated…

Let $X\to\mathbf{P}_{\Delta}$ and $X^{\vee}\to\mathbf{P}_{\nabla}$ be smooth MPCP desingularizations admitting the stated nef-partitions, and let $Y$ and $Y^{\vee}$ be the associat…

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…

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…

Katzarkov–Kontsevich–Pantev conjecture. One should have

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 $\frX \to \AA^1$ be a Tyurin degeneration to $\frZ_0 \cup_{\frD} \frZ_1$, with mirror Calabi-Yau $\widecheck{\frX_{gen}}$ and mirror divisor $\frD$. A Tyurin degeneration is a…

Let $X$ be a Calabi–Yau manifold. A Lagrangian fibration is a map $\pi:X\to B$ onto a topological manifold $B$ whose fibers are graded with respect to a holomorphic volume form…

Let $V$ be the Milnor fiber under consideration, let $W^\vee$ be its Berglund–Hübsch mirror polynomial, and let $\Gamma^\vee$ be the maximal symmetry group of $W^\vee$, … Write…

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…

Let $(X_t,\omega_{CY,t})$ be a polarised degeneration family of $n$-dimensional Calabi–Yau manifolds near the large complex structure limit, and let $\mu_t$ denote the normalised C…

Let $P(a_1,\dots,a_N\mid k_1,\dots,k_m)$ be a nonsingular complete intersection in weighted projective space. Let $t^p(x^*)$ be the mirror maps defined from the genus-zero virtual…

Let $g\geq 2$ be the genus, let ${\mathsf{r}}=2$ be the rank, and let ${\mathsf{d}}$ be the degree. Define the generating series $\check{\mathsf{QM}}^i_{\mathsf{d}}(q)$ and…

Let $U$ and $U^\vee$ be mirror $n$-dimensional log Calabi–Yau varieties. For a nonsingular quasi-projective variety $V$, let … be its perverse-mixed Hodge polynomial. Mirror P=W co…

Gross–Hacking–Keel mirror construction conjecture. The free $R$-module $V$ with basis $U^{\mathrm{trop}}(\mathbb Z)$ has a natural finitely generated $R$-algebra structure whose st…

SYZ conjecture. There is an affine manifold with singularities $B$ and maps $\phi:X\rightarrow B$ and $\check{\phi}:\check{X}\rightarrow B$ which are dual special Lagrangian fibrat…

Strong mirror symmetry conjecture. Every pair consisting of a smooth Fano manifold and a divisor on it has a toric Landau–Ginzburg model.

Let $X$ be a semi-Fano toric manifold, meaning that its anti-canonical divisor $-K_X$ is nef. Let $W^{\text{HV}}$ and $W^{\text{LF}}$ denote its Hori–Vafa and Lagrangian Floer supe…

Let $X$ and $\check{X}$ be Calabi–Yau manifolds which are mirror to each other. A special Lagrangian torus fibration is a fibration with torus fibers calibrated by the special Lagr…