Numerical spectrums control cohomological spectrums

Let X be a smooth irreducible projective variety over a field k with dimX=d. Let τ:Ql→C be any field embedding. Let f:X→X be a surjective endomorphism. We show that for every i=0,…,2d, the spectral radius of f∗ on the numerical group Ni(X)⊗R and on the l-adic cohomology group H2i(Xk‾,Ql)⊗C are the same. As a consequence, if f is q-polarized for some q>1, we show that the norm of every eigenvalue of f∗ on the jth cohomology group is qj∕2 for all j=0,…,2d. This generalizes Deligne’s theorem for Weil’s Riemann hypothesis to arbitary polarized endomorphisms and proves a conjecture of Tate. We also get some applications for the counting of fixed points and its “moving target” variant. In fact, we studied the more general actions of certain cohomological coorespondences, and we get the above results as consequences in the endomorphism setting.