123 problems

Castelnuovo bound conjecture. The invariant vanishes when

Let $M$ be a Calabi–Yau threefold, let $H^*(M)$ denote its cohomology, and let $W^+(M)$ be the algebra generated by $T_n(\lambda)$ for $n{\geqslant} 0$ and $\lambda\in H^*(M)$, sub…

Let $M$ be a Calabi–Yau threefold, and let $\beta$ be a nonzero curve class in $M$. The symbols $\operatorname{Z_{GW}^{\prime}}(M;u)_\beta$ and $\operatorname{Z_{PT}}(M;q)_\beta$ d…

The JMGS conjecture. The small $J$-function satisfies

Let $\mathcal{Y}\to B$ be the one-parameter family of smooth, multiply connected Calabi–Yau threefolds over … where $\xi$ is a primitive fifth root of unity, and write $Y_t$ for th…

The B-model of a Calabi–Yau threefold has a partition function associated with its monodromy action. B-model partition-function conjecture. The analogue of Theorem holds for the B-…

Let $T(Y,\overline{Y},\widetilde{Y})$ be a geometric transition satisfying a good condition, for example that its birational morphism $\phi$ contracts $k$ exceptional divisors to a…

Let $T(Y,\overline{Y},\widetilde{Y})$ be a geometric transition, and let $Y^{\circ}$ and $\widetilde{Y}^{\circ}$ be mirror partners of $Y$ and $\widetilde{Y}$, respectively. Morris…

Let $Y=(Y,J,h)$ be a general Calabi–Yau threefold and let $\chi\in\mathcal{K}_{\mathbb{C}}(Y)$ satisfy $\operatorname{Im}\chi=-\frac12\operatorname{Im}h$. Call $(Y,\chi)$ a polariz…

Let $X$ and $Y$ be Calabi–Yau threefolds, and write $D(X)$ and $D(Y)$ for their bounded derived categories. Derived-equivalence conjecture for Calabi–Yau threefolds. If … then $X$…

Let $(M,J,\omega,\Omega)$ be an (almost) Calabi–Yau $3$-fold with compact underlying $6$-manifold $M$, complex structure $J$, Kähler form $\omega$, and holomorphic volume form…

Fix $\mathbb K=\mathbb C$. Let $P$ be a Calabi–Yau $3$-fold, set $\mathcal A=\operatorname{coh}(P)$, and let $\mathfrak F_{\mathcal A}$ and $K(\mathcal A)$ be as in the source. Let…

Let $X$ be a tamed $K3$-fibered Calabi–Yau threefold, and let $n_C$ denote the virtual number of rational curves in the fiberwise class $C\in H_2(X,\mathbf Z)_f$. Suppose that the…

Let $X$ be a Calabi–Yau threefold, meaning that $K_X$ is trivial. For curve classes $\beta\ne0$, let $Z^{\mathrm{DT}}(X)'$ be the reduced Donaldson–Thomas partition function, and l…

Let $X$ be a Calabi–Yau $3$-fold and let $L$ be a nef divisor on $X$. Nef abundance conjecture. The divisor $L$ is semiample. The surrounding discussion derives this assertion from…

Let $X$ be the Calabi–Yau threefold considered above, let $\beta\in H_2(X,\mathbb Z)$, and let … be the surjective morphism from the normalization of the sheaf moduli space to the…

The modularity conjecture. Any rigid Calabi–Yau threefold $X$ defined over ${\mathbb Q}$ is modular: its $L$-series coincides, up to finitely many Euler factors, with the $L$-serie…

Hurwitz-number congruence conjecture. If $d$ is not divisible by $4$, $6$, or $9$, then

Let $X$ be a noncompact toric Calabi–Yau threefold with $\dim H_4(X,\mathbb{Z})=0$, and let $\{l^1,\dots,l^m\}$ define $X$ via symplectic quotient. Define … where … and … If…

Consider the A-model on a Calabi–Yau threefold $X$. Let $u_i$ be the logarithm of the B-model complex deformation parameter $z_i$ obtained from the toric construction of the mirror…

Let $X$ be a Calabi–Yau threefold, and let $\Delta$ be the discriminant locus in the moduli space of conformal field theories. Its primary component is the component corresponding t…

The chamber formula conjecture. In the fundamental chamber,

Modular mirror-symmetry identity.

Rationality conjecture. The function

Let $X_{\zeta_{+}}$ be any Calabi–Yau threefold constructed by the GLSM flop procedure, with coordinates $z_{1},\ldots,z_{r}$ and divisors $\widehat D_{\alpha-1,\alpha}$. Assume th…