Virtual classes of Artin stacks

Poma, Flavia

doi:10.1007/s00229-014-0694-6

Cited by 7 publications

(6 citation statements)

References 18 publications

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“…For Deligne-Mumford stacks this criterion is [BF97, Thm 4.5], and for algebraic stacks it is stated in [AP19, Cor 8.5] and [Pom15,Thm 3.5]. However, we emphasize a difference in wording: in place of condition (2b), the references cited require only that Def(f ) be a torsor under Ext 0 (Lf * E, I) (and automorphisms are isomorphic to Ext −1 (Lf * E, I)).…”

Section: Theorem 421 ([Ols06]mentioning

confidence: 99%

The Moduli of Sections Has a Canonical Obstruction Theory

Webb¹

2021

Preprint

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We give a detailed proof that locally Noetherian moduli stacks of sections carry canonical obstruction theories. As part of the argument we construct a dualizing sheaf and trace map, in the lisse-étale topology, for families of tame twisted curves, when the base stack is locally Noetherian. Contents 10 4. Obstruction theories via the Fundamental Theorem 17 Appendix A. Three descent theorems for lisse-étale sheaves on algebraic stacks 24 Appendix B. Functoriality of the Fundamental Theorem 30 References 40

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Section: Theorem 421 ([Ols06]mentioning

confidence: 99%

The Moduli of Sections Has a Canonical Obstruction Theory

Webb¹

2021

Preprint

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“…] is the vector bundle over [T * N/G] associated with g * , and µ is the section defined by the moment map. By standard constructions, the intrinsic normal cone C X embeds into the pullback vector bundle stack ι * E g * , which defines a perfect obstruction theory E • X on X, in the sense of [47]. Alternatively, E • X fits into the distinguished triangle…”

Section: Higgs Branch and Quasimapsmentioning

confidence: 99%

Virtual Coulomb branch and vertex functions

Zhou¹

2021

Preprint

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We introduce a variation of the K-theoretic quantized Coulomb branch constructed by Braverman-Finkelberg-Nakajima, by application of a new virtual intersection theory. In the abelian case, we define Verma modules for such virtual Coulomb branches, and relate them to the moduli spaces of quasimaps into the corresponding Higgs branches. The descendent vertex functions, defined by K-theoretic quasimap invariants of the Higgs branch, can be realized as the associated Whittaker functions. The quantum q-difference modules and Bethe algebras (analogue of quantum K-theory rings) can then be described in terms of the virtual Coulomb branch. As an application, we prove the wall-crossing result for quantum q-difference modules under the variation of GIT. Nonabelian cases are also treated via abelianization.

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“…If we truncate 2-stacks to 1-stacks (i.e., taking π ≤1 . ), then we have the version of virtual pullbacks in [31], which is similar to working with obstruction sheaves instead of vector bundle stacks. We hope to address these matters in [32].…”

Section: Deformation Spacesmentioning

confidence: 99%

“…When f is DM, L f ∈ D ≤0 qcoh (X), and a perfect obstruction theory φ : 4,31]), and any such imbedding corresponds to some POT. So a POT can be viewed either as some map in the derived category or an embedding of the intrinsic normal cone into some vector bundle stack.…”

Section: Gysin Pullbacks Commute With Proper Pushforwards and Flat Pu...mentioning

confidence: 99%