A vanishing theorem for Fano varieties in positive characteristic

2002/01/20 by Minhyong Kim, Kim, Minhyong
Mathematics · #11G25 #14G05 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11G25 #msc:14G05

paper · pdf · doi:10.48550/arxiv.math/0201183

A correct proof of the original vanishing theorem (mod torsion) has been given in the meanwhile by Esnault. This note strengthens that result slightly

openalex publication_date 2002/01/20 · arxiv created 2002/07/05 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a Kodaira-type vanishing theorem for the Witt vector sheaf on a Fano variety over a perfect field of characteristic p. As a corollary, we deduce that the number of rational points on a Fano variety over a finite field with q=pⁿ elements is congruent to 1 mod q. This refines a result of Esnault which say that the number is congruent to 1 mod p.

Related

• Congruences for the number of rational points, Hodge type and motivic conjectures for Fano varieties
• Varieties with too many rational points
• The failure of Kodaira vanishing for Fano varieties, and terminal singularities that are not Cohen-Macaulay
• The number of smooth varieties in an MMP on a 3-fold of Fano type
• Vanishing theorems of Kodaira type for Witt canonical sheaves