92 problems

Barannikov–Kontsevich conjecture. The Frobenius structure $(S',\eta_F|_{S'},\Phi_F|_{S'})$ should be a Frobenius submanifold of the Frobenius manifold constructed by Barannikov–Kon…

Landau–Ginzburg compactification conjecture. $X$ admits a Landau–Ginzburg model with a smooth log Calabi–Yau compactification.

Let $r$ be the spin parameter, let $g\geq 2$, let $F_g^{x^r,o}$ denote the open $r$-spin potential, and let $\phi_g$ denote the corresponding closed $r$-spin potential. The notatio…

Normed Calabi–Yau conjecture. For each Delzant polytope $\triangle$, the $\infty$-trace $\Theta$ on $MF(W_\triangle)$ is a normed Calabi–Yau structure.

Orbifold KKP conjecture. One should have

Katzarkov–Kontsevich–Pantev conjecture. One should have

Let $(W,\langle J\rangle)$ be an admissible LG pair, and let $I_{\rm FJRW}^{\rm sm}(t,-z)$ be its small FJRW $I$-function. Let $\tau(t)$ be defined by the coefficient of $z^0$ in t…

Let $W$ be a nondegenerate quasihomogeneous singularity of general type. Let $I_{\rm FJRW}^{\rm sm}(t,z)$ be the small FJRW $I$-function, let $\tau(t)$ be determined by the coeffic…

Dolgachev–Nikulin duality conjecture. There is an isomorphism of lattices

Let $Y$ be a smooth Fano variety of index $r$, let $D\subseteq Y$ be a smooth anti-canonical divisor, and let $X$ be the corresponding cyclic cover. A graded exact Lagrangian torus…

Let $Y$ be a smooth Fano variety of index $r$ and $D_Y\thinspace\subseteq\thinspace Y$ a smooth anti-canonical divisor. Let $X$ be the $r$-fold cyclic covering of $Y$ branched alon…

Let $X$ be a derived critical locus, and let $f$ range over potentials whose derived critical locus is $X$. Derived spectrum equivalence conjecture. The derived spectrum is the sam…

Let $X$ be a smooth variety with a smooth nef anticanonical divisor $D$, and consider the local Calabi–Yau manifold $\mathcal O_X(-D)$. The instanton corrections are the correction…

Let $Y$ be a Fano manifold with smooth anticanonical divisor $D$. For a suitable submonoid $P\subset \operatorname{Pic}(Y)$, let $(X,W)$ be the intrinsic mirror dual, where … is a…

Let $f$ be an invertible polynomial in $n$ variables, let $G\subset G_f$ be a group of diagonal symmetries, and let $S\subset S_n$ preserve $f$ and $G$. Let…

Let $f$ be an invertible polynomial in $n$ variables, let $G\subset G_f$ be a group of diagonal symmetries, and let $S\subset S_n$ preserve $f$ and $G$. Let…

Let $(Y,w)$ be a Landau–Ginzburg model admitting a tame compactification. Let $f^{p,q}(Y,w)$ be the Hodge-theoretic invariants defined from the Hodge filtration on relative cohomol…

Let $X$ be a smooth Fano variety of dimension $n$, and let $Y$ be a Calabi–Yau compactification of its toric Landau–Ginzburg model. Let $k_Y$ be the number of irreducible component…

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 $X$ be a quasi-Fano variety with a simple normal crossings anticanonical divisor $D$, and suppose that $X$ admits a semi-stable degeneration into $X_1\cup_{D_0}X_2$. The mirror…

Let $X$ be a quasi-Fano variety with a smooth anticanonical divisor $D$, and let $D_1+D_2$ be a simple normal crossings anticanonical divisor of $X$. Let $(X^\beth,h)$ denote the r…

Let $X_1$ and $X_2$ be Fano varieties with mirror Landau–Ginzburg models $(X_1^\vee,W_1)$ and $(X_2^\vee,W_2)$. Write $\pi_{W_i}(t)$ for the classical period of the model, and let…

Let $U$ and $V=U^\star$ be mirror log Calabi–Yau varieties with theta-function bases, and let $P\subset N_U$ and $Q=P^\star\subset M_U$ be dual reflexive polygons. The polytope $P$…

Let $\widecheck{\frZ}_0 \to \AA^1$ and $\widecheck{\frZ}_1 \to \AA^1$ be Landau–Ginzburg models, and suppose that the fibres of $\widecheck{\frZ}_0 \to \AA^1$ have one dimensional…

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…