2024/12/02 by Xie, Junyi
Computer Science · Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic and Geometric Analysis #FOS: Mathematics #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2412.01216
openalex publication_date 2024/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a smooth irreducible projective variety over a field k of dimension 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(X_k,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 j-th 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. Indeed we studied the more general actions of certain cohomological coorespondences and we get the above results as consequences in the endomorphism setting.