5 problems

The log-terminal anticanonical divisor conjecture. For every $n\in\mathbb{N}$, there exists a divisor $D\in |-nK_X|$ such that

Anticanonical dual-complex conjecture. The dual simplicial complex of an anticanonical simple normal crossings divisor should be a sphere, or a closely related space.

Let $X$ be a Kähler manifold with a smooth anticanonical divisor $D$, and let $\Omega$ be the holomorphic volume form on $X\setminus D$ with residue $\Omega_D$ along $D$. Relative…

The SYZ mirror identification conjecture. The subset $M_D$ is the Strominger–Yau–Zaslow mirror of $D$. This identifies the mirror of the anticanonical divisor with the locus where…

Let $X$ be a compact Kähler manifold, let $D$ be an anticanonical divisor, let $U$ be a neighborhood of $D$, and let $M$ be the moduli space of pairs $(L,\nabla)$ consisting of a s…