Motivic and $\ell$-adic realizations of the category of singularities of the zero locus of a global section of a vector bundle

TL;DR

This paper investigates the motivic and ll-adic realizations of the dg category of singularities associated with zero loci of sections of line bundles, providing formulas connecting these realizations to geometric and algebraic structures.

Contribution

It introduces a new approach to compute motivic and ll-adic realizations of singularity categories using formulas derived from the zero locus of line bundle sections.

Findings

Derived formulas for ll-adic realizations of singularity categories.

Connected motivic and ll-adic realizations with geometric structures.

Extended results to special fibers over regular local rings.

Abstract

We study the motivic and $\ell$-adic realizations of the dg category of singularities of the zero locus of a global section of a line bundle over a regular scheme. We will then use the formula obtained in this way together with a theorem due to D.~Orlov and J.~Burke - M.~Walker to give a formula for the $\ell$-adic realization of the dg category of singularities of the special fiber of a scheme over a regular local ring of dimension $n$.