Lifting vector bundles to Witt vector bundles

TL;DR

This paper investigates the conditions under which vector bundles on a scheme can be lifted to its Witt vector scheme, providing new insights into deformation theory and specific non-liftability results for Grassmannians.

Contribution

It offers new criteria for lifting vector bundles to Witt vector schemes, extends classical deformation results, and demonstrates non-liftability in specific geometric cases.

Findings

Line bundles can be lifted to Witt vector schemes.

The tautological bundle on certain Grassmannians cannot be lifted.

Connections to classical deformation theory and non-liftability results.

Abstract

Let $X$ be a scheme. Let $r \geq 2$ be an integer. Denote by $W_r(X)$ the scheme of Witt vectors of length $r$, built out of $X$. We are concerned with the question of extending (=lifting) vector bundles on $X$, to vector bundles on $W_r(X)$-promoting a systematic use of Witt modules and Witt vector bundles. To begin with, we investigate two elementary but significant cases, in which the answer to this question is positive: line bundles, and the tautological vector bundle of a projective bundle over an affine base. We then offer a simple (re)formulation of classical results in deformation theory of smooth varieties over a field $k$ of characteristic $p>0$, and extend them to reduced $k$-schemes. Some of these results were recently recovered, in another form, by Stefan Schr\"oer. As an application, we prove that the tautological vector bundle of the Grassmannian $Gr_{\mathbb{F}_p}(m,n)$ does not extend to $W_2(Gr_{\mathbb{F}_p}(m,n))$, if $2 \leq m \leq n-2$. To conclude, we establish a connection to the work of Zdanowicz, on non-liftability of some projective bundles.