92 problems

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 $(Y,w)$ 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…

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…

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

Let $(Y_1,w_1)$ and $(Y_2,w_2)$ be gauged Landau–Ginzburg models with $Y_i$ smooth, a group $G$ acting on each $Y_i$, and $w_i$ a section of $\operatorname{\mathcal O}(\chi)$ for a…

Let $W$ be an invertible polynomial with maximal diagonal symmetry group $G_W$, and let $(W^T,\{1\})$ be its Berglund–Hübsch–Krawitz mirror pair. Write $(H_W,\bullet)$ for the FJRW…

Let $Z_1,\ldots,Z_t$ be smooth varieties in projective spaces, each either Calabi–Yau or of general type, defined by the polynomials $w_j$ as in the source. Form the grading group…

Consider the Lagrangian cones $\mathcal L_{\operatorname{FJRW}}$ and $\mathcal L_{\operatorname{GW}}$ in the symplectic spaces $\mathcal V_{\operatorname{FJRW}}$ and…

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

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 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…