Frobenius pushforwards of of vector bundles on projective spaces

2024/02/12 by Feliks Rączka, Rączka, Feliks
Mathematics · #13A35 #14J60 #14M25 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2402.07554

openalex publication_date 2024/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate when the filtration induced by Beilinson's spectral sequence splits non-canonically into a direct sum decomposition. We conclude that for any vector bundle E on a projective space over an algebraically closed field of characteristic p>0 there exists r₀ such that for r≥ r₀ the Frobenius pushforward F^r_*E decomposes as a direct sum of line bundles and exterior powers of the cotangent bundle (we also give a variant for the "toric Frobenius map" valid in any characteristic). As an application we give a short proof of Klyachko's theorem for vanishing of the cohomology of toric vector bundles on projective spaces.

Related

• Decomposition of Frobenius pushforwards of line bundles on wonderful compactifications
• A note on the Frobenius morphism on toric varieties
• Rank one summands of Frobenius pushforwards of line bundles on G/P
• Harder-Narasimhan Filtrations which are not split by the Frobenius maps
• Frobenius pull backs of vector bundles in higher dimensions