Grothendieck-Lefschetz for vector bundles

TL;DR

This paper proves Dao's conjecture that vector bundles on punctured spectra of complete intersection rings are free under certain depth conditions, extending the Grothendieck-Lefschetz theorem to higher rank bundles.

Contribution

The authors use deformation theory to settle Dao's conjecture, providing a criterion for when vector bundles are trivial on punctured spectra of complete intersections.

Findings

Dao's conjecture is confirmed using deformation theoretic methods.

Examples demonstrate the sharpness of the conjecture's assumptions.

Results have implications for splitting vector bundles on projective space.

Abstract

According to the Grothendieck-Lefschetz theorem from SGA 2, there are no nontrivial line bundles on the punctured spectrum $U_R$ of a local ring $R$ that is a complete intersection of dimension $\ge 4$. Dao conjectured a generalization for vector bundles $\mathscr{V}$ of arbitrary rank on $U_R$: such a $\mathscr{V}$ is free if and only if $\mathrm{depth}_R(\mathrm{End}_R(\Gamma(U_R, \mathscr{V}))) \ge 4$. We use deformation theoretic techniques to settle Dao's conjecture. We also present examples showing that its assumptions are sharp and draw consequences for splitting of vector bundles on complete intersections in projective space.