# 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 Jul 2 '20 at 15:48
• @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)? Jul 3 '20 at 1:14
• 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. Jul 3 '20 at 20:47