Let $X$ be a smooth complex projective variety (of dimension two or higher) and $D=\bigcup D_i$ be a simple normal crossings divisor on $X$ such that:
- $D$ is ample (maybe we need very ample but I am not sure)
- $D$ is toric in the sense that each $D_i$ is toric, and the intersections $D_{i_1}\cap \dotsb \cap D_{i_k}$ are toric faces of $D_{i_1},\dotsc,D_{i_k}$
- $(D-D_i)\cap D_i =\partial D_i$ (i.e., every codimension one face of $D_i$ belongs to some $D_i\cap D_j$);
Is it true/false that $(X,D)$ is log Calabi–Yau?
Note that, by the 3rd assumption and adjunction $(K_X+D)\rvert_{D}$ is trivial. So I feel Lefschetz Theorem (together with assumption one) should imply that $(X,D)$ is log Calabi–Yau!
As a non-toric example, let $X$ be a cubic surface and $D$ be a triangle of lines (there are 45 of them on a generic cubic surface). Then $D$ is a hyperplane section of $X\subset \mathbb{P}^3$ and thus $(X,D)$ is log CY.
Rmk. We probably need extra assumptions on the dual complex of $D$; see https://arxiv.org/pdf/1503.08320.pdf
