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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

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    $\begingroup$ What if $X$ is a hyperbolic curve and $D$ is a single point? $\endgroup$ Jan 18, 2023 at 12:27
  • $\begingroup$ Are there higher dimensional examples? $\endgroup$ Jan 18, 2023 at 13:27
  • $\begingroup$ In the context I was thinking about, $X-D$ has hyperkahler metric, but I doubt the last condition is really important. $\endgroup$ Jan 18, 2023 at 13:36
  • $\begingroup$ There are higher-dimensional examples: let $X_1,\dots,X_d$ be hyperbolic curves, let $D_1,\dots,D_r$ be singleton divisors in each $X_i$, let $X$ be $X_1\times \dots \times X_r$, and let $D$ be the sum of the pullbacks of the $D_i$ by the coordinate projections. $\endgroup$ Jan 18, 2023 at 17:22
  • $\begingroup$ That does not meet the toric condition on $D_i$! $\endgroup$ Jan 18, 2023 at 18:52

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