The Uniqueness Theorem for Gysin Coherent Characteristic Classes of Singular Spaces

TL;DR

This paper introduces a recursive geometric scheme for computing characteristic classes of singular algebraic varieties, demonstrating its effectiveness through applications to Schubert varieties and the Goresky-MacPherson L-class.

Contribution

It develops a systematic, recursive computational framework for characteristic classes of singular spaces satisfying a Gysin axiom, with explicit geometric methods and applications to Schubert varieties.

Findings

Established a recursive scheme for characteristic class computation.

Applied the scheme to singular Schubert varieties without Poincaré duality.

Derived a uniqueness theorem for Gysin coherent characteristic classes.

Abstract

We establish a general computational scheme designed for a systematic computation of characteristic classes of singular complex algebraic varieties that satisfy a Gysin axiom in a transverse setup. This scheme is explicitly geometric and of a recursive nature terminating on genera of explicit characteristic subvarieties that we construct. It enables us e.g. to apply intersection theory of Schubert varieties to obtain a uniqueness result for such characteristic classes in the homology of an ambient Grassmannian. Our framework applies in particular to the Goresky-MacPherson L-class by virtue of the Gysin restriction formula obtained by the first author in previous work. We illustrate our approach for a systematic computation of the L-class in terms of normally nonsingular expansions in examples of singular Schubert varieties that do not satisfy Poincar\'{e} duality over the rationals.