2016/11/03 by Tuyen Trung Truong, Truong, Tuyen Trung · 2 citations
Chemistry · Mathematics · Pharmacology, Toxicology and Pharmaceutics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Alkaloids: synthesis and pharmacology #Chromatography in Natural Products #Dynamical Systems (math.DS) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1611.01124
openalex publication_date 2016/11/03 · openalex created_date 2016/11/11 · openalex updated_date 2026/07/28
Let \mathbbK be an algebraically closed field, X a smooth projective variety over \mathbbK and f:X→ X a dominant regular morphism. Let Ni(X) be the group of algebraic cycles modulo numerical equivalence. Let χ(f) be the spectral radius of the pullback f^*:H^*(X,ℚl)→ H^*(X,ℚl) on l-adic cohomology groups, and λ(f) the spectral radius of the pullback f^*:N^*(X)→ N^*(X). We prove in this paper, by using consequences of Deligne's proof of Weil's Riemann hypothesis, that χ(f)=λ(f). This answers affirmatively a question posed by Esnault and Srinivas. Consequently, the algebraic entropy log χ(f) of an endomorphism is both a birational invariant and étale invariant. More general results are proven if either \mathbbK=\mathbbFp or the Fundamental Conjecture D (numerical equivalence vs homological equivalence) holds. Among other results in the paper, we show that if some properties of dynamical degrees, known in the case \mathbbK=ℂ, hold in positive characteristics, then simple proofs of Weil's Riemann hypothesis follow.