Motivic Chern classes of configuration spaces

Comparison of motivic Chern classes and stable envelopes for cotangent bundles

“…To study the intersection with hyperplane $$H$$ we use the limit technique with respect to the torus $$\sigma _H$$. We use the definition of limit map from [25, Definition 4.1]. It is a map defined on a subring of the localized $$K$$‐theory: $$\begin{equation*} \lim _{{\bf t}\rightarrow 0}: S_A^{-1}K^A(pt)[y,y^{-1}] \dashrightarrow S_{A/\sigma _H}^{-1}K^{A/\sigma _H}(pt)[y,y^{-1}]\,.$$…”

“…To study the intersection with hyperplane 𝐻 we use the limit technique with respect to the torus 𝜎 𝐻 . We use the definition of limit map from [25,Definition 4.1]. It is a map defined on a subring of the localized 𝐾-theory:…”

“…[12, Theorem 5.11.7]) and the above properties. For examples of computations, see [13, 16, 25]. In [16] (see also [15]) it was found that for $$G$$‐equivariant varieties with a finite number of orbits, the motivic Chern classes $$mC_{{y}}^G$$ of $$G$$‐orbits satisfy axioms similar to those of the stable envelopes.…”

“…Our main technical tool is the study of limits of Laurent polynomials (of one or many variables). The limit technique was investigated for motivic Chern classes [16,25,48] as well as for stable envelopes [34,45]. We use it mainly to prove various containments of Newton polytopes.…”

“…It induces an isomorphism 𝐾 𝐴 (𝑝𝑡) ≃ 𝐾 𝐴∕𝜎 𝐻 (𝑝𝑡)[𝐭, 𝐭−1 ]. Then the limit map is defined on the subring 𝐾 𝐴∕𝜎 𝐻 (𝑝𝑡)[𝐭][𝑦, 𝑦−1 ] by killing all positive powers of t. technical details and extension to the localized 𝐾-theory see[25, Section 4].Let 𝐻 be a hyperplane corresponding to a facet of the polytope 𝑁 𝐴 (𝑒𝑢( ν− 𝐹 ′ )) ⊂ 𝐸 𝐻 . Thus (for a more detailed discussion, see Remark 6.5) Let F ⊂ 𝑋 𝜎 𝐻 be a component of the fixed point set which contains 𝐹 ′ .Proposition 4.3 and [25,…”