Kawaguchi-Silverman Conjecture for Certain Surjective Endomorphisms

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 surjec-tive 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 en-domorphism. The main ingredients are the equivariant minimal model program, the effectiveness of the anti-canonical divisor and a charac-terization of toric pairs.