Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Virtual signed Euler characteristics

Authors: Yunfeng Jiang and Richard P. Thomas
Journal: J. Algebraic Geom. 26 (2017), 379-397
Published electronically: October 21, 2016
Full-text PDF

Abstract | References | Additional Information

Abstract: Roughly speaking, to any space $ M$ with perfect obstruction theory we associate a space $ N$ with symmetric perfect obstruction theory. It is a cone over $ M$ given by the dual of the obstruction sheaf of $ M$ and contains $ M$ as its zero section. It is locally the critical locus of a function.

More precisely, in the language of derived algebraic geometry, to any quasi-smooth space $ M$ we associate its $ (\!-\!1)$-shifted cotangent bundle $ N$.

By localising from $ N$ to its $ \mathbb{C}^*$-fixed locus $ M$ this gives five notions of a virtual signed Euler characteristic of $ M$:

  1. The Ciocan-Fontanine-Kapranov/Fantechi-Göttsche signed virtual Euler characteristic of $ M$ defined using its own obstruction theory,
  2. Graber-Pandharipande's virtual Atiyah-Bott localisation of the virtual cycle of $ N$ to $ M$,
  3. Behrend's Kai-weighted Euler characteristic localisation of the virtual cycle of $ N$ to $ M$,
  4. Kiem-Li's cosection localisation of the virtual cycle of $ N$ to $ M$,
  5. $ (-1)^{\textrm {vd}}$ times by the topological Euler characteristic of $ M$.
Our main result is that (1)=(2) and (3)=(4)=(5). The first two are deformation invariant while the last three are not.

References [Enhancements On Off] (What's this?)

  • [Be] Kai Behrend, Donaldson-Thomas type invariants via microlocal geometry, Ann. of Math. (2) 170 (2009), no. 3, 1307-1338. MR 2600874,
  • [BF] K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45-88. MR 1437495,
  • [CKL] H.-L. Chang, Y.-H. Kiem and J. Li, Torus localization and wall crossing for cosection localized virtual cycles, arXiv:1502.00078.
  • [CL] Huai-Liang Chang and Jun Li, Gromov-Witten invariants of stable maps with fields, Int. Math. Res. Not. IMRN 18 (2012), 4163-4217. MR 2975379,
  • [CK] Ionuţ Ciocan-Fontanine and Mikhail Kapranov, Virtual fundamental classes via dg-manifolds, Geom. Topol. 13 (2009), no. 3, 1779-1804. MR 2496057,
  • [Co] Kevin Costello, Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4, Pure Appl. Math. Q. 9 (2013), no. 1, 73-165. MR 3126501,
  • [Da] B. Davison, The critical CoHA of a self dual quiver with potential, arXiv:1311.7172.
  • [FG] Barbara Fantechi and Lothar Göttsche, Riemann-Roch theorems and elliptic genus for virtually smooth schemes, Geom. Topol. 14 (2010), no. 1, 83-115. MR 2578301,
  • [GP] T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487-518. MR 1666787,
  • [HL] Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010. MR 2665168
  • [Ill] Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 (French). MR 0491680
  • [Ka] Masaki Kashiwara, Index theorem for constructible sheaves, Astérisque 130 (1985), 193-209. Differential systems and singularities (Luminy, 1983). MR 804053
  • [KL] Young-Hoon Kiem and Jun Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26 (2013), no. 4, 1025-1050. MR 3073883,
  • [MT] D. Maulik and D. Treumann, Constructible functions and Lagrangian cycles on orbifolds, arXiv:1110.3866.
  • [PTVV] Tony Pantev, Bertrand Toën, Michel Vaquié, and Gabriele Vezzosi, Shifted symplectic structures, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 271-328. MR 3090262,
  • [Sch] Timo Schürg, Deriving Deligne-Mumford stacks with perfect obstruction theories, Geom. Topol. 17 (2013), no. 1, 73-92. MR 3035324,
  • [TVa] Bertrand Toën and Michel Vaquié, Moduli of objects in dg-categories, Ann. Sci. École Norm. Sup. (4) 40 (2007), no. 3, 387-444 (English, with English and French summaries). MR 2493386,
  • [TVe] Bertrand Toën and Gabriele Vezzosi, Homotopical algebraic geometry. II. Geometric stacks and applications, Mem. Amer. Math. Soc. 193 (2008), no. 902, x+224. MR 2394633,
  • [VW] Cumrun Vafa and Edward Witten, A strong coupling test of $ S$-duality, Nuclear Phys. B 431 (1994), no. 1-2, 3-77. MR 1305096,

Additional Information

Yunfeng Jiang
Affiliation: Department of Mathematics, University of Kansas, 405 Jayhawk Boulevard, Lawrence, Kansas 66045

Richard P. Thomas
Affiliation: Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

Received by editor(s): September 2, 2014
Published electronically: October 21, 2016
Article copyright: © Copyright 2016 University Press, Inc.

American Mathematical Society