Kawaguchi–Silverman conjecture for certain surjective endomorphisms

  • Sheng Meng

    School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Road, Shanghai 200241, People's Republic of China; Max-Planck-Institut für Mathematik, Vivatsgasse 7, Bonn 53111, Germany
  • De-Qi Zhang

    Department of Mathematics, National University of Singapore, Singapore 119076, Republic of Singapore
Kawaguchi–Silverman conjecture for certain surjective endomorphisms cover
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Abstract

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 -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.

Cite this article

Sheng Meng, De-Qi Zhang, Kawaguchi–Silverman conjecture for certain surjective endomorphisms. Doc. Math. 27 (2022), pp. 1605–1642

DOI 10.4171/DM/X13