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Building blocks of étale endomorphisms of complex projective manifolds. (English) Zbl 1185.14012

The paper under review studies étale endomorphisms of non-singular projective complex varieties. The approach is interesting and motivated by a standard subdivision of varieties according to the sign of the Kodaira dimension. The statements of theorems are too technical to appear in a review. I will try to summarize the ideas and give a hint at the flavour of the paper.
Fix a nonsingular projective variety \(X\) and an étale endomorphism \(f\).
If \(X\) is uniruled, the authors associate an endomorphism \(h\) of a non-uniruled normal variety \(Y\). This can even be improved, assuming standard minimal model program conjectures, to an étale endomorphism of a smooth variety of non negative Kodaira dimension.
If \(\kappa(X)>0\) then one can attach to \(f\) an étale endomorphism of the general fiber of a Iitaka fibration of \(X\) [see also K. Ueno, Classification theory of algebraic varieties and compact complex spaces. Notes written in collaboration with P. Cherenack. Lecture Notes in Mathematics 439. Berlin-Heidelberg-New York: Springer-Verlag (1975; Zbl 0299.14007)].
If \(\kappa(X)=0\) then one can attach endomorphisms to weak CY varieties and abelian varieties.
Summing up all one can see the endomorphisms of weak CY and abelian varieties as the building blocks of all endomorphisms of projective varieties. It should be said that going backward (that is, recovering \(f\) from the attached endomorphisms) is quite complicated and not always possible [see Y. Fujimoto and N. Nakayama, J. Math. Kyoto Univ. 47, No. 1, 79–114 (2007; Zbl 1138.14023) and Y. Fujimoto, Publ. Res. Inst. Math. Sci. 38, No. 1, 33–92 (2002; Zbl 1053.14049)].

MSC:

14E20 Coverings in algebraic geometry
14E07 Birational automorphisms, Cremona group and generalizations
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
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