A ‘Darboux theorem’ for shifted symplectic structures on derived Artin stacks, with applications

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

doi:10.2140/gt.2015.19.1287

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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

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

(b) If $(X, ω_{X})$ is a $(- 1)$ –shifted symplectic derived Artin stack and $X = t_{0} (X)$ 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 ${\overset{̌}{P}}_{X, s}^{∙}$ on $X$ , such that whenever $T$ is a scheme and $t : T \to X$ is smooth of relative dimension $n$ , $T$ is locally modelled on a critical locus $Crit (f : U \to A^{1})$ , and $t^{*} ({\overset{̌}{P}}_{X, s}^{∙}) [n]$ is modelled on the perverse sheaf of vanishing cycles ${P V}_{U, f}^{∙}$ of $f$ .

(d) If $(X, s)$ is a finite-type oriented d–critical stack, we can define a natural motive ${M F}_{X, s}$ in a ring of motives ${\bar{ℳ}}_{X}^{st, \hat{μ}}$ on $X$ , such that if $T$ is a scheme and $t : T \to X$ is smooth of dimension $n$ , then $T$ is modelled on a critical locus $Crit (f : U \to A^{1})$ , and $L^{- n ∕ 2} ⊙ t^{*} ({M F}_{X, s})$ is modelled on the motivic vanishing cycle ${M F}_{U, f}^{mot, ϕ}$ 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/

Volume 19, issue 3 (2015)