6 problems

Kontsevich–Soibelman conjecture. In the limit, both mirror dual manifolds $X$ and $X^\vee$ become fiber bundles with toroidal fibers over the same base $\mathscr{Y}$, which is a Mo…

For each fixed complex dimension, consider compact hyper-Kähler manifolds and identify two such manifolds when they are homeomorphic, equivalently when they have the same topologic…

Let $N$ be a hyper-Kähler manifold of complex dimension $2n$. An isometric deformation of its complex structure is a deformation preserving a chosen Calabi–Yau metric. The SYZ defo…

A compact hyper-Kähler manifold is a compact Kähler manifold with a holomorphic symplectic form; its complex dimension is even. The SYZ finiteness conjecture. In each fixed complex…

Master-equation conjecture. For SYZ fibrations on log Calabi–Yau varieties near the tropical limit, there exists a model of $\mathfrak{C}$ for which $\mathfrak{m}_0$ is an element…

Weak family unobstructedness conjecture. Weak family unobstructedness should arise in SYZ mirror symmetry from the existence of a degeneration of the complex structure on $X$ to th…