When is the self-intersection of a subvariety a fibration?

TL;DR

The paper establishes a precise criterion for when the derived self-intersection of a smooth subvariety forms a fibration, leading to a generalized Hochschild-Kostant-Rosenberg isomorphism and connections to various mathematical theories.

Contribution

It provides a necessary and sufficient condition for derived self-intersections to be fibrations, extending classical isomorphisms and linking to multiple areas in geometry and topology.

Findings

Characterization of when derived self-intersection is a fibration

Derivation of a generalized HKR isomorphism

Connections to path spaces, formality, and Lie and symplectic geometry

Abstract

We provide a necessary and sufficient condition for the derived self-intersection of a smooth subscheme inside a smooth scheme to be a fibration over the subscheme. As a consequence we deduce a generalized HKR isomorphism. We also investigate the relationship of our result to path spaces in homotopy theory, Buchweitz-Flenner formality in algebraic geometry, and draw parallels with similar results in Lie theory and symplectic geometry.