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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
∈ X we can find a
‘minimal’ smooth atlas
φ : U
→ 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 P ̌ X , 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
→ A 1 ) ,
and
t ∗ ( 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
ℳ ¯ X st , μ ̂ 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
→ 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
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