Abstract
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.
Citation
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
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