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2015 A ‘Darboux theorem’ for shifted symplectic structures on derived Artin stacks, with applications
Oren Ben-Bassat, Christopher Brav, Vittoria Bussi, Dominic Joyce
Geom. Topol. 19(3): 1287-1359 (2015). DOI: 10.2140/gt.2015.19.1287

Abstract

This is the fifth in a series of papers on the ‘k–shifted symplectic derived algebraic geometry’ of Pantev, Toën, Vaquié and Vezzosi. We extend our earlier results from (derived) schemes to (derived) Artin stacks. We prove four main results:

(a) If (X,ωX) is a k–shifted symplectic derived Artin stack for k < 0, then near each x X we can find a ‘minimal’ smooth atlas φ: U X, such that (U,φ(ωX)) may be written explicitly in coordinates in a standard ‘Darboux form’.

(b) If (X,ωX) is a (1)–shifted symplectic derived Artin stack and X = t0(X) the classical Artin stack, then X extends to a ‘d–critical stack’ (X,s), as by Joyce.

(c) If (X,s) is an oriented d–critical stack, we define a natural perverse sheaf P̌X,s on X, such that whenever T is a scheme and t: T X is smooth of relative dimension n, T is locally modelled on a critical locus Crit(f : U A1), and t(P̌X,s)[n] is modelled on the perverse sheaf of vanishing cycles PVU,f of f.

(d) If (X,s) is a finite-type oriented d–critical stack, we can define a natural motive MFX,s in a ring of motives ¯Xst,μ̂ on X, such that if T is a scheme and t: T X is smooth of dimension n, then T is modelled on a critical locus Crit(f : U A1), and Ln2 t(MFX,s) is modelled on the motivic vanishing cycle MFU,fmot,ϕ of f.

Our results have applications to categorified and motivic extensions of Donaldson–Thomas theory of Calabi–Yau 3–folds.

Citation

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Oren Ben-Bassat. Christopher Brav. Vittoria Bussi. Dominic Joyce. "A ‘Darboux theorem’ for shifted symplectic structures on derived Artin stacks, with applications." Geom. Topol. 19 (3) 1287 - 1359, 2015. https://doi.org/10.2140/gt.2015.19.1287

Information

Received: 4 December 2013; Revised: 4 April 2014; Accepted: 8 June 2014; Published: 2015
First available in Project Euclid: 16 November 2017

zbMATH: 1349.14003
MathSciNet: MR3352237
Digital Object Identifier: 10.2140/gt.2015.19.1287

Subjects:
Primary: 14A20
Secondary: 14D23, 14F05, 14N35, 32S30

Rights: Copyright © 2015 Mathematical Sciences Publishers

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