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

Ben-Bassat, Oren¹ ; Brav, Christopher² ; Bussi, Vittoria¹ ; Joyce, Dominic¹

¹ The Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
² Institute for Advanced Study, Princeton University, Einstein Drive, Princeton, NJ 08540, USA

Geometry & topology, Tome 19 (2015) no. 3, pp. 1287-1359.

Résumé

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

(b) If $\left(X,{\omega }_X\right)$ is a $\left(-1\right)$–shifted symplectic derived Artin stack and $X=t_0\left(X\right)$ the classical Artin stack, then $X$ extends to a ‘d–critical stack’ $\left(X,s\right)$, as by Joyce.

(c) If $\left(X,s\right)$ is an oriented d–critical stack, we define a natural perverse sheaf ${\check{P}}_{X,s}^{\bullet }$ 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 $Crit\left(f:U\to {\mathbb{A}}^1\right)$, and $t^{\ast }\left({\check{P}}_{X,s}^{\bullet }\right)\left[n\right]$ is modelled on the perverse sheaf of vanishing cycles ${\mathcal{P}\mathcal{V}}_{U,f}^{\bullet }$ of $f$.

(d) If $\left(X,s\right)$ is a finite-type oriented d–critical stack, we can define a natural motive ${MF}_{X,s}$ in a ring of motives ${\overline{\mathcal{ℳ}}}_X^{st,\widehat{\mu }}$ 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 $Crit\left(f:U\to {\mathbb{A}}^1\right)$, and ${\mathbb{L}}^{-n∕2}\odot t^{\ast }\left({MF}_{X,s}\right)$ is modelled on the motivic vanishing cycle ${MF}_{U,f}^{mot,\varphi }$ of $f$.

Our results have applications to categorified and motivic extensions of Donaldson–Thomas theory of Calabi–Yau $3$–folds.

Détail

DOI : 10.2140/gt.2015.19.1287

Classification : 14A20, 14F05, 14D23, 14N35, 32S30
Keywords: derived algebraic geometry, derived stack, shifted symplectic structure, perverse sheaf, vanishing cycles, motivic invariant, Calabi–Yau manifold, Donaldson–Thomas theory

Bibliographie

[1] A A Beĭlinson, J Bernstein, P Deligne, Faisceaux pervers, from: "Analysis and topology on singular spaces, I", Astérisque 100, Soc. Math. France (1982) 5

[2] E Bouaziz, I Grojnowski, A $d\!$–shifted Darboux theorem,

[3] C Brav, V Bussi, D Dupont, D Joyce, B Szendroi, Symmetries and stabilization for sheaves of vanishing cycles,

[4] C Brav, V Bussi, D Joyce, A ‘Darboux theorem’ for derived schemes with shifted symplectic structure,

[5] C Brav, V Bussi, S Meinhardt, On motivic vanishing cycles of critical loci,

[6] J Denef, F Loeser, Geometry on arc spaces of algebraic varieties, from: "European Congress of Math., Vol. I" (editors C Casacuberta, R M Miró-Roig, J Verdera, S Xambó-Descamps), Progr. Math. 201, Birkhäuser (2001) 327

[7] A Dimca, Sheaves in topology, Springer (2004)

[8] T Ekedahl, On the adic formalism, from: "The Grothendieck Festschrift, Vol. II" (editors P Cartier, L Illusie, N M Katz, G Laumon, K A Ribet), Progr. Math. 87, Birkhäuser (1990) 197

[9] E Freitag, R Kiehl, Étale cohomology and the Weil conjecture, Ergeb. Math. Grenzgeb. 13, Springer (1988)

[10] D Gaitsgory, N Rozenblyum, Crystals and $\mathscr{D}$–modules,

[11] R Hartshorne, Algebraic geometry, Graduate Texts in Math. 52, Springer (1977)

[12] H Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, I, II, Ann. of Math. 79 (1964) 205

[13] D Joyce, Motivic invariants of Artin stacks and ‘stack functions’, Q. J. Math. 58 (2007) 345

[14] D Joyce, A classical model for derived critical loci,

[15] D Joyce, Y Song, A theory of generalized Donaldson–Thomas invariants, Mem. Amer. Math. Soc. 1020, Amer. Math. Soc. (2012)

[16] R Kiehl, R Weissauer, Weil conjectures, perverse sheaves and $l$–adic Fourier transform, Ergeb. Math. Grenzgeb. 42, Springer (2001)

[17] M Kontsevich, Y Soibelman, Stability structures, motivic Donaldson–Thomas invariants and cluster transformations,

[18] M Kontsevich, Y Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants, Commun. Number Theory Phys. 5 (2011) 231

[19] A Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999) 495

[20] Y Laszlo, M Olsson, The six operations for sheaves on Artin stacks, I: Finite coefficients, Publ. Math. Inst. Hautes Études Sci. 107 (2008) 109

[21] Y Laszlo, M Olsson, The six operations for sheaves on Artin stacks, II: Adic coefficients, Publ. Math. Inst. Hautes Études Sci. 107 (2008) 169

[22] Y Laszlo, M Olsson, Perverse $t$–structure on Artin stacks, Math. Z. 261 (2009) 737

[23] G Laumon, L Moret-Bailly, Champs algébriques, Ergeb. Math. Grenzgeb. 39, Springer (2000)

[24] Y Liu, W Zheng, Enhanced six operations and base change theorem for Artin stacks,

[25] Y Liu, W Zheng, Enhanced adic formalism, biduality and perverse $t$–structures for higher Artin stacks, preprint (2012)

[26] E Looijenga, Motivic measures, from: "Séminaire Bourbaki, \rm1999/2000", Astérisque 276 (2002) 267

[27] M Olsson, Sheaves on Artin stacks, J. Reine Angew. Math. 603 (2007) 55

[28] T Pantev, B Toën, M Vaquié, G Vezzosi, Shifted symplectic structures, Publ. Math. Inst. Hautes Études Sci. 117 (2013) 271

[29] A G M Paulin, The Riemann–Hilbert correspondence for algebraic stacks,

[30] B Toën, Higher and derived stacks: a global overview, from: "Algebraic geometry" (editors D Abramovich, A Bertram, L Katzarkov, R Pandharipande, M Thaddeus), Proc. Sympos. Pure Math. 80, Amer. Math. Soc. (2009) 435

[31] B Toën, G Vezzosi, Homotopical algebraic geometry, II: Geometric stacks and applications, Mem. Amer. Math. Soc. 902, Amer. Math. Soc. (2008)