Chern character, semi-regularity map and obstructions

TL;DR

This paper uses Chern characters to connect local Hilbert functors with Hochschild homology, providing new insights into semi-regularity maps and their role in obstructing deformations of subvarieties.

Contribution

It constructs a natural transformation linking Hilbert functors to Hochschild homology, offering a novel proof of Bloch's theorem on semi-regularity and obstructions.

Findings

Reconstruction of semi-regularity map via Hochschild homology

New proof that semi-regularity map kills certain obstructions

Enhanced understanding of deformation theory for subvarieties

Abstract

Using Chern character, we construct a natural transformation from the local Hilbert functor to a functor of Artin rings defined from Hochschild homology, which allows us to reconstruct the semi-regularity map and the infinitesimal Abel-Jacobi map. Combining that construction of the semi-regularity map with obstruction theory of functors of Artin rings, we give a different proof of a theorem of Bloch stating that the semi-regularity map annihilates certain obstructions to embedded deformations of a closed subvariety which is a locally complete intersection.