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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
A ‘Darboux theorem’ for shifted symplectic structures on derived Artin stacks, with applications

Oren Ben-Bassat, Christopher Brav, Vittoria Bussi and Dominic Joyce

Geometry & Topology 19 (2015) 1287–1359
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.

Keywords
derived algebraic geometry, derived stack, shifted symplectic structure, perverse sheaf, vanishing cycles, motivic invariant, Calabi–Yau manifold, Donaldson–Thomas theory
Mathematical Subject Classification 2010
Primary: 14A20
Secondary: 14F05, 14D23, 14N35, 32S30
References
Publication
Received: 4 December 2013
Revised: 4 April 2014
Accepted: 8 June 2014
Published: 21 May 2015
Proposed: Richard Thomas
Seconded: Jim Bryan, Ciprian Manolescu
Authors
Oren Ben-Bassat
The Mathematical Institute
University of Oxford
Radcliffe Observatory Quarter
Woodstock Road
Oxford OX2 6GG
UK
http://https://sites.google.com/site/orenbenbassat/
Christopher Brav
Institute for Advanced Study
Princeton University
Einstein Drive
Princeton, NJ 08540
USA
Vittoria Bussi
The Mathematical Institute
University of Oxford
Radcliffe Observatory Quarter
Woodstock Road
Oxford OX2 6GG
UK
Dominic Joyce
The Mathematical Institute
University of Oxford
Radcliffe Observatory Quarter
Woodstock Road
Oxford OX2 6GG
UK
http://people.maths.ox.ac.uk/~joyce/