31 problems

Let $Q$ be a connected component of the coarse moduli space of triples $(X,E=E_1+\dots+E_n,\Theta)$ such that $X$ is a connected smooth projective complex variety,…

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…

The Calabi conjecture. There is a unique bi-polyhedral Kähler affine structure on $\Sigma\backslash D$ of type $(\lambda,\nu)$ such that the metric is Monge–Ampère:

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 $\Gamma\subset\operatorname{SU}(3)$, 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…

Calabi–Yau critical-value conjecture. If $c^1(X)=0$, then

Let $X\subset\mathbb{P}^{n+1}(\mathbb{C})$ be a general hypersurface of degree $n+2$, and let $\Gamma\in A^{2n}(X\times X\times X)$ be the cycle defined in the source. Voisin's con…

Let $X\subset\mathbb{P}^{n+1}(\mathbb{C})$ be a smooth hypersurface. Assume that either the dimension $n$ is odd, or the degree of $X$ is $n+2$, so that $X$ is Calabi–Yau. Intersec…

Rational-curves density conjecture. The union of rational curves in $X'$ is Zariski dense in $X'$.

Rational-curves conjecture for torsion-canonical manifolds. The manifold $X$ is an étale quotient of an Abelian variety.

Unobstructedness conjecture. The deformations of both restricted log maps are unobstructed for all $a\in H_1$ and $b\in H_2$.

Let the moduli space of a Calabi–Yau threefold be equipped with its Weil–Petersson geometry, and consider its geodesics. Stability conjecture. All Weil–Petersson geodesics on the m…

Let $V$ be a compact Calabi–Yau manifold, and let $Z\subset V$ be a smooth ample divisor. Set $U=V\setminus Z$, which is a Weinstein manifold. Let $\mathrm{Fuk}^{w}(U)$ be the grad…

Let $X$ be a Calabi–Yau threefold. An even-dimensional homology class $[\Sigma]$ is called large when, for example, $[\Sigma]=N[\Sigma_0]$ with $N\gg1$. If the associated black bra…

Non-BPS attractor equidistribution conjecture. The non-BPS attractors equidistribute according to $\rho$; more precisely,

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…

Abundance conjecture. $\mathcal L$ is semiample, i.e. there exists a positive integer $m$ such that $\mathcal L^{\otimes m}$ is generated by global sections.