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

Abstract

We study the motivic and $$\ell $$ ℓ -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 $$\ell $$ ℓ -adic realization of the dg category of singularities of the zero locus of a global section of a vector bundle. In particular, we obtain a formula for the $$\ell $$ ℓ -adic realization of the dg category of singularities of the special fiber of a scheme over a regular local ring of dimension n.

Singularity categories of normal crossings surfaces, descent, and mirror symmetry

“…In this section we shall study the descent properties of DSing for schemes and MF for twisted Landau-Ginzburg models. The results are not entirely new and can be found in some form in the work of Preygel [Pre12] (for the untwisted case), Blanc-Robalo-Toën-Vezzosi [BRTV18], and Pippi [Pip22]. However, our presentation differs from those references be emphasizing the notion of 1-affineness.…”

“…We also include some discussion of the theory of singular support for coherent sheaves, since we have found this concept very helpful in visualizing the geometric constructions we shall make. Some references that take a point of view close to the one presented here include Preygel [Pre12], Arinkin-Gaitsgory [AG15], Blanc-Robalo-Toën-Vezzosi [BRTV18], and Pippi [Pip22]. Of course the general theory of derived categories of singularities is much older.…”

“…Relative singularity categories have been studied by Efimov-Positselski [EP15], Blanc-Robalo-Toën-Vezzosi [BRTV18], and Pippi [Pip22], and they arise most naturally when considering Landau-Ginzburg models whose total space is itself singular. Let f : Y → A 1 be a flat morphism and let X = f −1 (0) be the zero fiber; we do not assume that Y is regular.…”

“…The perfect relative singularity category is the version studied in [BRTV18,Pip22]. In this paper the term "relative singularity category" always refers to the first, coherent version.…”