Grothendieck-Lefschetz for vector bundles
Kestutis Cesnavicius

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 of a local ring that is a complete intersection of dimension . Dao conjectured a generalization for vector bundles of arbitrary rank on : such a is free if and only if . 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.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Homotopy and Cohomology in Algebraic Topology · Commutative Algebra and Its Applications
