Quadric invariants and degeneration in smooth-etale cohomology

TL;DR

This paper studies how cohomological invariants of quadric bundles behave under degeneration in algebraic geometry, extending topological results to an algebraic setting using stacks and advanced cohomological tools.

Contribution

It provides a formula for the Gysin boundary map of cohomological invariants of quadric bundles in algebraic geometry, generalizing previous topological results to algebraic stacks.

Findings

Derived a formula for the boundary behavior of characteristic classes in algebraic geometry.

Extended topological invariants to algebraic stacks using algebraic stacks and cohomology.

Results apply to smooth-etale cohomology of quadric bundles over algebraic stacks.

Abstract

For a regular pair $(X,Y)$ of schemes of pure codimension 1 on which 2 is invertible, we consider quadric bundles on $X$ which are nondegenerate on $X-Y$, but are minimally degenerate on $Y$. We give a formula for the behaviour of the cohomological invariants (characteristic classes) of the nondegenerate quadric bundle on $X-Y$ under the Gysin boundary map to the etale cohomology of $Y$ with mod 2 coefficients. The results here are the algebro-geometric analogs of topological results for complex bundles proved earlier by Holla and Nitsure, continuing further the algebraization program which was commenced with a recent paper by Bhaumik. We use algebraic stacks and their smooth-etale cohomologies, $A^1$-homotopies and Gabber's absolute purity theorem as algebraic replacements for the topological methods used earlier, such as CW complexes, real homotopies, Riemannian metrics and tubular neighbourhoods. Our results also hold in smooth-etale cohomology for quadric bundles over algebraic stacks on which 2 is invertible.