18 problems

Let $X\rightarrow X_0\leadsto Z$ be a conifold transition, and let $\check{X}$ and $\check{Z}$ be mirror Calabi-Yau threefolds to $X$ and $Z$, respectively. Morrison's conjecture.…

A conifold transition of a smooth Calabi–Yau 3-fold $X$ is a flat degeneration to a nodal Calabi–Yau 3-fold followed by a small resolution. If the mirror Calabi–Yau 3-folds $Y^*$ a…

Conifold transition conjecture. All families of Calabi--Yau threefolds can be joined by a sequence of conifold transitions.

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 conifold transition, meaning a geometric transition whose singular intermediate variety has only ordinary double points. Reverse conifold…

Let $X_0$ be a conifold degeneration of a generic Calabi–Yau complete intersection $X\subset G(k,n)$, let $p$ be the number of nodes of $X_0$, and set $\alpha=(k-1)(n-k-1)$. Let…

Let $chat{X}$ be a Calabi–Yau threefold, and let … be a conifold transition. Let $(\Omega_t,\omega_t,h_t)$ denote the holomorphic volume form, Hermitian form, and bundle metric inv…

Let $(g,H)$ be a pair consisting of a balanced metric $g$ and a Hermitian–Yang–Mills metric $H$ satisfying … where $\omega=i g_{j\bar{k}}\,dz^j\wedge d\bar{z}^k$. Suppose that the…

Let $\widehat{X}$ be a Kähler Calabi–Yau threefold. A conifold transition is a process … where a birational contraction maps holomorphic curves to ordinary double points of the ana…

Conifold-transition connectivity conjecture. All simply connected Calabi–Yau threefolds can be joined by birational contraction followed by smoothing.

Betti-number conjecture.

Betti-number conjecture.

Related-nodes conjecture. A set of nodes $p_1,\ldots,p_k$ in $(B,\mathcal P,\phi)$ can be simultaneously resolved by the above process if and only if they are related.

Castaño-Bernard–Matessi conjecture. The existence of tropical $2$-cycles containing the nodes should imply that the tropical conifold can be resolved and its dual tropical conifold…

A Calabi–Yau 3-fold is a six-real-dimensional Calabi–Yau space. Green–Hübsch's conjecture. Topologically distinct Calabi–Yau spaces are connected to each other by means of conifold…

Small-transition decomposition conjecture. Every small geometric transition which does not admit any conifold degeneration is equivalent to a composition of conifold transitions. C…

A geometric transition is abbreviated g.t.; a geometric transition is small and rigid when it has those properties. A factorization of a geometric transition $T$ is an expression ……

Type I conifold-degeneration conjecture. Every type I geometric transition admits a conifold degeneration. In particular, the canonical model of a type I geometric transition is a…