If $A$ is an abelian variety over a field, the Kummer variety $K\_A$ associated to $A$ is obtained as the quotient of $A$ by the involution $\iota: a \mapsto -a$. If $A$ is a surface, it is well-known that a resolution of $K\_A$ can be constructed simply by blowing up the image (under the quotient map) of the set of $2$-torsion points. Is this statement still true if $A$ has dimension at least $3$? Precisely, is the variety obtained from $K\_A$ by blowing up the image of the locus of $2$-torsion points under the quotient map always smooth?



Resolutions of higher-dimensional Kummer varieties
Yes, it is smooth. What is different from dim 2, is that the resolution is not crepant, so the resolution of $K\_A$ is not a Calabi--Yau variety.


