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Frobenius pushforwards of of vector bundles on projective spaces

Feliks Rączka

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 $\mathcal{E}$ on a projective space over an algebraically closed field of characteristic $p>0$ there exists $r_{0}$ such that for $r\geq r_{0}$ the Frobenius pushforward $\mathsf{F}^{r}_{*}\mathcal{E}$ 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.

Frobenius pushforwards of of vector bundles on projective spaces

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 on a projective space over an algebraically closed field of characteristic there exists such that for the Frobenius pushforward 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.
Paper Structure (5 sections, 13 theorems, 40 equations, 1 figure)

This paper contains 5 sections, 13 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['Thm2']}
  4. Proof of Theorem \ref{['MainThm']}
  5. Theorem of Klyachko

Key Result

Theorem 1.1

Let $k$ be an algebraically closed field of arbitrary characteristic and let ${\mathcal{E}}$ be a vector bundle on ${\mathbb P}^{n}_{k}$. Then for $m\geq m({\mathcal{E}})$ there exists a decomposition Moreover, $a_{i}=\dim_{k}H^{i}({\mathbb P}^{n}_{k},{\mathcal{E}}).$

Figures (1)

  • Figure 1: This graphic illustrates why Lemma \ref{['DaggerLemma1']} holds. Beilinson's spectral sequence is concentrated in the square $\{(r,s):-r,s\in\{0,1,\dots,n\}\}$. If ${\mathcal{E}}$ has property ($\dagger$) then there are no nontrivial arrows between elements with $1\leq s\leq n-1$. In particular any entry above (resp. below) the diagonal admits only nonzero arrows from (resp. to) the elements in the top (resp. bottom) row. Circled elements on the diagonal have to satisfy $E^{-r,r}_{1}=E^{-r,r}_{\infty}$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Bott's Formula, Okonek
  • Lemma 2.2: Beilinson, Beilinson
  • Theorem 2.3: Beilinson's spectral sequence, Huybrechts
  • Theorem 2.4: Horrocks' criterion, Okonek
  • Lemma 2.5
  • proof
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['DaggerLemma1']}
  • ...and 15 more