𝐾-theory of line bundles and smooth varieties

Abstract

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