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We prove Kawaguchi-Silverman conjecture (KSC) and Shibata's conjecture onample canonical heights for endomorphisms on several classes of algebraicvarieties including varieties of Fano type and projective toric varieties. Wealso prove KSC for group endomorphisms of linear algebraic groups. We alsopropose a possible approach to the conjecture using equivariant MMP.

Kawaguchi-Silverman conjecture for endomorphisms on several classes of varieties