K-theory of line bundles and smooth varieties

TL;DR

This paper establishes a K-theoretic criterion for smoothness of quasi-projective varieties, showing that certain K-theory groups' equality involving an ample line bundle characterizes smoothness.

Contribution

It introduces a new K-theoretic condition involving line bundles that characterizes the smoothness of quasi-projective varieties.

Findings

K_q(X) = K_q(ℓ) for all q ≤ dim(X)+1 implies smoothness

Provides a practical criterion for smoothness based on K-theory

Connects line bundle properties with the smoothness of varieties

Abstract

We give a $K$-theoretic criterion for a quasi-projective variety to be smooth. If $\mathbb{L}$ is a line bundle corresponding to an ample invertible sheaf on $X$, it suffices that $K_q(X) = K_q(\mathbb{L})$ for all $q\le\dim(X)+1$.