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Let $U$ be a complex affine log Calabi-Yau variety, which I take to mean a smooth affine variety which admits a compactification in a smooth projective variety $X$ with an snc anti-canonical compactifying divisor $D$ (this seems to be a bit more stringent than what is sometimes meant by log CY).

Suppose $U$ is diffeomorphic to $(\mathbb{C}^*)^n$. Is it known that $U$ is algebraically an affine torus? I'm also curious about the same question with "diffeomorphic" replaced by "homotopy equivalent."

I should add that if $\operatorname{dim}(U)=2$, I'm pretty sure one could prove the answer is ``yes" just by classification.

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  • $\begingroup$ Maybe numdam.org/article/CM_1981__43_2_253_0.pdf answers your question? It seems to me that we have a quasi-Albanese morphism $a:U\to A$ where $\dim A=\bar q(U)=\dim U$. By Theorem 28, $a$ is an open algebraic fiber space and by Corollary 29 it is birational $\endgroup$ – Hacon 3 hours ago

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