On the dynamical and arithmetic degreesof rational self-maps of algebraic varieties

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>:</m:mo> <m:mi>X</m:mi> <m:mo>⤏</m:mo> <m:mi>X</m:mi> </m:mrow> </m:math> f:X X be a dominant rational map of a smooth projective variety defined over a characteristic 0 global field K , let δ f be the dynamical degree of f , and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>h</m:mi> <m:mi>X</m:mi> </m:msub> <m:mo>:</m:mo> <m:mi>X</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mover accent="true"> <m:mi>K</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mo>)</m:mo> </m:mrow> <m:mo>→</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>∞</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> hX:X(K̄)→ [1,∞ ) be a Weil height relative to an ample divisor. We prove that for every <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ϵ</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> ε >0 there is a height bound <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>h</m:mi> <m:mi>X</m:mi> </m:msub> <m:mo>∘</m:mo> <m:msup> <m:mi>f</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>≪</m:mo> <m:msup> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>δ</m:mi> <m:mi>f</m:mi> </m:msub> <m:mo>+</m:mo> <m:mi>ϵ</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mi>n</m:mi> </m:msup> <m:msub> <m:mi>h</m:mi> <m:mi>X</m:mi> </m:msub> <m:mo>,</m:mo> </m:mrow> </m:math> hX∘ fⁿ (δ f+ε )ⁿ hX, valid for all points whose f -orbit is well-defined, where the implied constant depends only on X , h X , f , and ϵ. An immediate corollary is a fundamental inequality <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mover> <m:mi>α</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mi>f</m:mi> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mi>P</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>≤</m:mo> <m:msub> <m:mi>δ</m:mi> <m:mi>f</m:mi> </m:msub> </m:mrow> </m:math> α ̄f(P)≤ δ f for the upper arithmetic degree. If further f is a morphism and D is a divisor satisfying an algebraic equivalence <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>f</m:mi> <m:mo>*</m:mo> </m:msup> <m:mi>D</m:mi> <m:mo>≡</m:mo> <m:mi>β</m:mi> <m:mi>D</m:mi> </m:mrow> </m:math> f^*D≡ β D for some <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>β</m:mi> <m:mo>></m:mo> <m:msqrt> <m:msub> <m:mi>δ</m:mi> <m:mi>f</m:mi> </m:msub> </m:msqrt> </m:mrow> </m:math> β >√δ f , we prove that the canonical height <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mover accent="true"> <m:mi>h</m:mi>