Harder-Narasimhan Filtrations which are not split by the Frobenius maps

TL;DR

This paper constructs explicit counterexamples to the higher-dimensional analogue of a known Frobenius splitting property for vector bundles, using algebraic geometry over integers and finite fields.

Contribution

It provides a family of vector bundles on smooth projective varieties over $ ext{Spec}( extbf{Z})$ that serve as counterexamples in positive characteristic, extending previous results.

Findings

Counterexamples exist over a dense set of primes

Construction uses Borel-Weil-Bott and Frobenius splitting techniques

Counterexamples are on homogeneous spaces associated with semisimple groups

Abstract

Let $X$ be a smooth projective variety over a perfect field $k$ of characteristic $p>0$, and $V$ be a vector bundle over $X$. It is well known that if $X$ is a curve and $V$ is not strongly semistable, then some Frobenius pullback $(F^t)^*V$ is a direct sum of strongly semistable bundles. A natural question to ask is whether this still holds in higher dimension. Indranil Biswas, Yogish I. Holla, A.J. Parameswaran, and S. Subramanian showed that there is always a counterexample to this over any algebraically closed field of positive characteristic which is uncountable. However, we will produce a smooth projective variety over $\mathbb Z$ and a rank 2 vector bundle on it, which, restricted to each prime $p$ in a nonempty open subset of $\spec\mathbb Z$, constitutes a counterexample over $p$. Indeed, given any split semisimple simply connected algebraic group $G$ of semisimple rank $>1$ over $\mathbb Z$, we will show that there exists a smooth projective homogeneous space $X_Z$ over $\mathbb Z$ and a vector bundle $V$ on $X_Z$ of rank 2 such that for each prime $p$ in a nonempty open subset of $\spec\mathbb Z$, the restriction $V\otimes\mathbb F_p$ as a vector bundle over $X_Z\otimes\mathbb F_p$ is a counterexample. We only use the Borel-Weil-Bott theorem in characteristic 0 and Frobenius Splitting of $G/B$ in characteristic $p$.