# Log Calabi-Yau variety diffeomorphic to an algebraic torus

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.

• 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 – Hacon 3 hours ago