The Hochschild-Kostant-Rosenberg isomorphism for quantized analytic cycles

TL;DR

This paper develops a detailed construction of generalized Hochschild-Kostant-Rosenberg (HKR) isomorphisms for smooth analytic cycles with split conormal sequences, addressing uniqueness and obstructions in complex geometry.

Contribution

It provides a rigorous account of Kashiwara's construction for generalized HKR isomorphisms, solving a recent problem on their uniqueness and introducing a cycle class obstruction.

Findings

Solved the problem of uniqueness of HKR isomorphisms for the diagonal injection.

Constructed a cycle class as an obstruction for cycles to be vanishing loci.

Extended HKR isomorphism theory to analytic cycles with infinitesimal retractions.

Abstract

In this article, we provide a detailed account of a construction sketched by Kashiwara in an unpublished manuscript concerning generalized HKR isomorphisms for smooth analytic cycles whose conormal exact sequence splits. It enables us, among other applications, to solve a problem raised recently by Arinkin and C\u{a}ld\u{a}raru about uniqueness of such HKR isomorphisms in the case of the diagonal injection. Using this construction, we also associate with any smooth analytic cycle endowed with an infinitesimal retraction a cycle class which is an obstruction for the cycle to be the vanishing locus of a transverse section of a holomorphic vector bundle.