67 problems

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

Moore's attractor conjecture. Attractor points are defined over $\overline{\mathbb{Q}}$. Moore verified this in several cases. The conjecture asks for algebraicity of the special p…

Let $H$ be a connected Hopf algebra of finite Gelfand–Kirillov dimension, with antipode $S$. The Nakayama automorphism conjecture. The Nakayama automorphism of the coalgebra $H$ is…

Consider the graph whose nodes are deformation classes of simply connected Calabi–Yau threefolds and whose edges are given by geometric transitions. Call a node primitive if its de…

Let $X$ be a smooth elliptic Calabi–Yau $4$-fold, and consider its vertical four-cycles, meaning four-cycles arising from the vertical part of the elliptic fibration. Their interse…

Let $X^{(n)}$ be a smooth Calabi–Yau $n$-fold, and let $\hat{D}$ be a divisor on $X^{(n)}$. The divisor $\hat{D}$ is nef if it has nonnegative intersection with every curve on…

Gepner's Fermat-quintic conjecture. The Gepner-model $N=2$ superconformal field theory should be isomorphic to the quantum nonlinear $\sigma$-model with the Fermat quintic threefol…

Dixon–Harvey–Vafa–Witten conjecture. The Euler characteristic of $Y$ should satisfy

Mirror-family conjecture. The pair $M_y+W_y$ is the mirror family of $X_A+Y_A$, with $M_y$ and $W_y$ forming different parts of the B-model moduli space and $X_A$ and $Y_A$ forming…

Singular-locus resemblance conjecture. The singular locus of the generalized special Lagrangian torus fibration should resemble $\tilde{\Gamma}$.

Let $X$ be a Calabi–Yau Gorenstein orbifold, and let a desingularization of $X$ be given. An inner local system is a compatible assignment of a flat orbifold line bundle to each se…

Calabi–Yau B-model conjecture. The generating function $\tilde{L}^{k,k}_n(e^x)$ gives information about the B-model of the mirror manifold of $M_k^k$. The paper presents this as an…

Calabi–Yau mirror-recursion conjecture. Formal iteration of the laws of Conjecture 2 down to $N=k$ should give

Cubic Calabi–Yau recursion conjecture. The recursive law is

Let $X$ be a Calabi–Yau threefold with $h^{1,1}(X)\neq h^{1,1}_{\mathrm{toric}}(X)$, and set … Let $\chi$ be a character depending on the defining equation of a singular locus. Tor…

Let $X$ be a $d$-dimensional Calabi–Yau variety whose associated modular form has weight $d+1$ on some $\Gamma_0(N)$, and write the level as … where the product is over primes of b…

Let $R_{\mathbf{v}}(t)$ be a contribution to the zeta function, and let the degree degeneration measure the decrease in $\operatorname{deg}(R_{\mathbf{v}})$ caused by a singularity…

Let $R_{\mathbf{v}}(t)$ be a contribution to the zeta function associated with a monomial class $\mathbf{v}$, and let $\operatorname{deg}(R_{\mathbf{v}})$ denote its rational-funct…

Singularity-degeneration conjecture. The degeneration of $\operatorname{deg}(R_{\mathbf{v}})$ for each piece of the zeta function of a Calabi–Yau manifold at a singular point in mo…

Let a family of Calabi–Yau hypersurfaces have Hodge numbers $h^{2,1}$ and $h^{2,1}_{\mathrm{poly}}$, and let $R_{\mathrm{excep}}(t)$ denote a factor of the numerator of its zeta fu…

Let $X$ be a Calabi–Yau manifold, and consider its category of A-branes, namely the derived category of the Fukaya category. The diagonal brane is the diagonal submanifold…

Let $\Gamma\subset\operatorname{SU}(3)$, let $X_\zeta$ be the moduli space associated with a generic parameter $\zeta$, and let $(M^{0,*,\Gamma},\bar{\partial}_\alpha)$ be the diff…

Let $Y$ be a Calabi–Yau manifold with $h^{2,0}(Y)=0$, and let $\pi:\mathcal{Y}\to S$ be a family of complex structures on $Y$ whose Kodaira–Spencer map is an isomorphism everywhere…

Suppose that $X$ is a Calabi–Yau manifold with $h^{2,0}(X)=0$, endowed with a complex structure satisfying the cone conjecture. Let … and let…

Let $X$ be a Calabi–Yau manifold with a chosen complex structure and suppose that $h^{2,0}(X)=0$. Let … let $\mathcal{K}$ be the Kähler cone of $X$, and let $\mathcal{K}_+$ be the…