An inequality on polarized endomorphisms

Abstract

AbstractWe show that assuming the standard conjectures, for any smooth projective varietyXof dimensionnover an algebraically closed field, there is a constant$$c>0$$c>0such that for any positive rational numberrand any polarized endomorphismfofX, we have$$\begin{aligned} \Vert G_r \circ f \Vert \le c \deg (G_r \circ f), \end{aligned}$$‖Gr∘f‖≤cdeg(Gr∘f),where$$G_r$$Gris a correspondence ofXso that for each$$0\le i\le 2n$$0≤i≤2n, its pullback action on thei-th Weil cohomology group is the multiplication-by-$$r^i$$rimap. This inequality is known to imply the generalized Weil Riemann hypothesis and is a special case of a more general conjecture by the authors’ work Hu and Truong (A dynamical approach to generalized Weil’s Riemann hypothesis and semisimplicity.arXiv:2102.04405v3, 2021).