Geometry & Topology

Deriving Deligne–Mumford stacks with perfect obstruction theories

Timo Schürg

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We show that every n–connective quasi-coherent obstruction theory on a Deligne–Mumford stack comes from the structure of a connective spectral Deligne–Mumford stack on the underlying topos. Working over a base ring containing the rationals, we obtain the corresponding result for derived Deligne–Mumford stacks.

Article information

Geom. Topol., Volume 17, Number 1 (2013), 73-92.

Received: 22 March 2012
Revised: 7 June 2012
Accepted: 8 September 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14A20: Generalizations (algebraic spaces, stacks) 18G55: Homotopical algebra
Secondary: 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.)

perfect obstruction theory derived moduli space


Schürg, Timo. Deriving Deligne–Mumford stacks with perfect obstruction theories. Geom. Topol. 17 (2013), no. 1, 73--92. doi:10.2140/gt.2013.17.73.

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  • K Behrend, B Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997) 45–88
  • M Kontsevich, Enumeration of rational curves via torus actions, from: “The moduli space of curves”, (R Dijkgraaf, C Faber, G van der Geer, editors), Progr. Math. 129, Birkhäuser, Boston, MA (1995) 335–368
  • J Li, G Tian, Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998) 119–174
  • J Lurie, Derived algebraic geometry, PhD thesis, Massachusetts Institute of Technology (2004)
  • J Lurie, A survey of elliptic cohomology (2007) Available at \setbox0\makeatletter\@url {\unhbox0
  • J Lurie, Quasi-coherent sheaves and Tannaka duality theorems (2011) Available at \setbox0\makeatletter\@url {\unhbox0
  • J Lurie, Spectral schemes (2011) Available at \setbox0\makeatletter\@url {\unhbox0
  • J Lurie, Higher algebra (2012) Available at \setbox0\makeatletter\@url {\unhbox0
  • B Toën, Higher and derived stacks: a global overview, from: “Algebraic geometry–-Seattle 2005. Part 1”, (D Abramovich, A Bertram, L Katzarkov, R Pandharipande, M Thaddeus, editors), Proc. Sympos. Pure Math. 80, Amer. Math. Soc. (2009) 435–487
  • B Toën, G Vezzosi, From HAG to DAG: derived moduli stacks, from: “Axiomatic, enriched and motivic homotopy theory”, (J P C Greenlees, editor), NATO Sci. Ser. II Math. Phys. Chem. 131, Kluwer Acad. Publ., Dordrecht (2004) 173–216
  • B Toën, G Vezzosi, Homotopical algebraic geometry, II: geometric stacks and applications, Mem. Amer. Math. Soc. 193 (2008) x+224