Let $A$ be an abelian variety over $\mathbb{C}$ and let $X\subset A$ be a closed subvariety. Let $X\to Y$ be the Ueno fibration. (That is, $Y$ is of general type and a closed subvariety of $A/B$ where $B$ is some abelian subvariety of $A$. Also, the morphism $X\to Y\to A/B$ equals the morphism $X\to A\to A/B$.)



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> > Is $X\to Y$ smooth?
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> > Are **all** fibres of $X\to Y$ abelian varieties?
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> > Is $X\to Y$ a torsor under some abelian scheme?
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Is the Ueno fibration smooth?
abx answered the question:


"Yes to the 3 questions. This is the content of Ueno's Theorem 10.9 in Classification theory of algebraic varieties and compact complex manifolds, LNM 439"


