Étale endomorphisms of 3-folds, III

Abstract

Up to a finite étale covering, we classify a smooth projective 3-fold X with $\mathrm{\kappa }(X)=-\mathrm{\infty }$ admitting a nonisomorphic étale endomorphism. We mainly study the case where there exist an FESP $Y_{\bullet }$ constructed from X by a sequence of blowing-downs of an ESP and an extremal ray $R_{\bullet }$ of $\overline{NE}(Y_{\bullet })$ such that the contraction morphisms associated to $R_{\bullet }$ give $Y_{\bullet }$ a smooth del Pezzo fiber space over an elliptic curve.