Degrees of iterates of rational maps on normal projective varieties

Let X be a normal projective variety defined over an algebraically closed field of arbitrary characteristic. We study the sequence of intermediate degrees of the iterates of a dominant rational selfmap of X, recovering former results by Dinh and Sibony (Ann. Sci. Éc. Norm. Supér. (4) 37 (2004) 959–971), and by Truong (J. Reine Angew. Math. 758 (2020) 139–182). Precisely, we give a new proof of the submultiplicativity properties of these degrees and of their birational invariance. Our approach exploits intensively positivity properties in the space of numerical cycles of arbitrary codimension. In particular, we prove an algebraic version of an inequality first obtained by Xiao (Ann. Inst. Fourier (Grenoble) 65 (2015) 1367–1369) and Popovici (Math. Ann. 364 (2016) 649–655), which generalizes Siu's inequality (see Trapani, Math. Z 219 (1995) 387–401) to algebraic cycles of arbitrary codimension. This allows us to show that the degree of a map is controlled up to a uniform constant by the norm of its action by pull‐back on the space of numerical classes in X.