The paper under review deals with the following question:
Let \(\{f_n\colon X_n\to X_{n+1}\}_{n=1,2,\dots}\) be an infinite tower of nonisomorphic finite étale coverings between smooth projective \(k\)-folds \(X_n\)’s with \(0\leq\kappa(X_n)=a<k\). It is true that for every \(n\), a suitable étale covering \(\widetilde X_n\to X_n\) has the structure of a smooth abelian scheme over a nonsingular projective variety \(W_n\) with \(0\leq\dim(W_n)<k\)?
The author shows that the answer to this question is affirmative if \((k,a)=(3,2)\) (in general), or if \((k,a)=(3,0)\) (under the assumption that all \(X_n\)’s are birationally isomorphic). In the first case, \(\widetilde X_n\cong W_n\times E_n\), with \(E_n\) an elliptic curve and \(W_n\) a surface of general type. In the second case \(\widetilde X_n\) can be: an abelian threefold, or a product \(S_n\times E_n\), with \(E_n\) an elliptic curve and \(S_n\) a surface which is birationally isomorphic to an abelian, or a \(K3\) surface.