This is the fifth in a series of papers on the
‘–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
is a
–shifted symplectic derived
Artin stack for
, then
near each
we can find a
‘minimal’ smooth atlas
,
such that
may be written explicitly in coordinates in a standard ‘Darboux form’.
(b) If
is a
–shifted symplectic derived
Artin stack and
the classical
Artin stack, then
extends
to a ‘d–critical stack’
,
as by Joyce.
(c) If
is an oriented d–critical stack, we define a natural perverse
sheaf on
, such that whenever
is a scheme and
is smooth of
relative dimension
,
is locally modelled
on a critical locus
,
and
is modelled on the perverse sheaf of vanishing cycles
of .
(d) If
is a finite-type oriented d–critical stack, we can define a natural motive
in a ring of
motives
on
, such that if
is a scheme and
is smooth of
dimension
, then
is modelled on a
critical locus
, and
is modelled on the
motivic vanishing cycle
of .
Our results have applications to categorified and motivic
extensions of Donaldson–Thomas theory of Calabi–Yau
–folds.