Geodesic
Kawaguchi–Silverman conjecture for certain surjective endomorphisms
Documenta mathematica, Tome 27 (2022), pp. 1605-1642

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We prove the Kawaguchi–Silverman conjecture (KSC), about the equality of arithmetic degree and dynamical degree, for every surjective endomorphism of any (possibly singular) projective surface. In high dimensions, we show that KSC holds for every surjective endomorphism of any Q-factorial Kawamata log terminal projective variety admitting one int-amplified endomorphism, provided that KSC holds for any surjective endomorphism with the ramification divisor being totally invariant and irreducible. In particular, we show that KSC holds for every surjective endomorphism of any rationally connected smooth projective threefold admitting one int-amplified endomorphism. The main ingredients are the equivariant minimal model program, the effectiveness of the anti-canonical divisor and a characterization of toric pairs.
DOI : 10.4171/dm/x13
Classification : 08A35, 14E30, 14J50, 37P55
Mots-clés : toric variety, Kawaguchi-Silverman conjecture, equivariant minimal model program, int-amplified endomorphism, arithmetic degree, dynamical degree 
Sheng Meng; De-Qi Zhang. Kawaguchi–Silverman conjecture for certain surjective endomorphisms. Documenta mathematica, Tome 27 (2022), pp. 1605-1642. doi: 10.4171/dm/x13
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     author = {Sheng Meng and De-Qi Zhang},
     title = {Kawaguchi{\textendash}Silverman conjecture for certain surjective endomorphisms},
     journal = {Documenta mathematica},
     pages = {1605--1642},
     year = {2022},
     volume = {27},
     doi = {10.4171/dm/x13},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/x13/}
}
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