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
    Jul 2 '20 at 15:48
  • $\begingroup$ @Hacon Sorry for being dense. I can see that this is suggestive, but how does one actually conclude the result --- how do we get that $a$ is actually an isomorphism (as opposed to just birational)? $\endgroup$ Jul 3 '20 at 1:14
  • $\begingroup$ Oh, I think this follows by the properties of the quasi-Albanese mmorphism. Since $q(X)=0$, then $A=(\mathbb C^*)^n$. Since $a:U\to A$ is the quasi-Albanese morphism, by the universal property of the quasi-Albenese ((1) of page 271 of Kawamata's paper applied to $\beta = id _U$) there is a morphism $f:A\to U$ such that $id_U=f\circ a$, hence $a$ is an isomorphism. $\endgroup$
    – Hacon
    Jul 3 '20 at 20:47

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