The Michigan Mathematical Journal

Note on MacPherson’s Local Euler Obstruction

Yunfeng Jiang

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This is a note on MacPherson’s local Euler obstruction, which plays an important role recently in the Donaldson–Thomas theory by the work of Behrend.

We introduce MacPherson’s original definition and prove that it is equivalent to the algebraic definition used by Behrend, following the method of González-Sprinberg. We also give a formula of the local Euler obstruction in terms of Lagrangian intersections. As an application, we consider a scheme or DM stack X admitting a symmetric obstruction theory. Furthermore, we assume that there is a C action on X that makes the obstruction theory C-equivariant. The C-action on the obstruction theory naturally gives rise to a cosection map in the Kiem–Li sense. We prove that Behrend’s weighted Euler characteristic of X is the same as the Kiem–Li localized invariant of X by the C-action.

Article information

Michigan Math. J., Volume 68, Issue 2 (2019), 227-250.

Received: 11 April 2017
Revised: 3 July 2017
First available in Project Euclid: 30 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 14A20: Generalizations (algebraic spaces, stacks)


Jiang, Yunfeng. Note on MacPherson’s Local Euler Obstruction. Michigan Math. J. 68 (2019), no. 2, 227--250. doi:10.1307/mmj/1548817530.

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  • [1] P. Aluffi, Limits of Chow groups, and a new construction of Chern–Schwartz–MacPherson classes, Pure Appl. Math. Q. 2 (2006), no. 4, 915–941.
  • [2] K. Behrend, Donaldson–Thomas type invariant via microlocal geometry, Ann. of Math. 170 (2009), no. 3, 1307–1338.
  • [3] K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. (1997), 1337–1398.
  • [4] K. Behrend and B. Fantechi, Symmetric obstruction theories and Hilbert scheme of point on threefolds, Algebra Number Theory 2 (2008), no. 3, 313–345.
  • [5] W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 2, second edition, Springer-Verlag, Berlin, 1998.
  • [6] V. Ginzburg, Characteristic varieties and vanishing cycles, Invent. Math. 84 (1986), 327–402.
  • [7] G. González-Sprinberg, L’obstruction locale d’Euler et le théorème de MacPherson, Astérisque 82–83 (1981), 7–32.
  • [8] Y. Jiang, The Pro-Chern–Schwartz–MacPherson class for DM stacks, Pure Appl. Math. Q. 11 (2015), no. 1, 87–114. arXiv:1412.3724.
  • [9] Y. Jiang and R. P. Thomas, Virtual signed Euler characteristics, J. Algebraic Geom. 26 (2017), 379–397.
  • [10] M. Kashiwara and P. Schapira, Sheaves on manifolds, Grundlehren Math. Wiss., 292, Springer-Verlag, Berlin, 1990.
  • [11] G. Kennedy, Macpherson’s Chern classes of singular algebraic varieties, Comm. Algebra 18:9 (1990), 2821–2839.
  • [12] Y.-H. Kiem and J. Li, Localizing virtual cycles by cosection, J. Amer. Math. Soc. 26 (2013), 1025–1050.
  • [13] J. Li and G. Tian, The virtual fundamental class for algebraic varieties, J. Amer. Math. Soc. (1998).
  • [14] R. MacPherson, Chern class for singular algebraic varieties, Ann. of Math. 100 (1974), no. 2, 423–432.
  • [15] D. Maulik and D. Treumann, Constructible functions and Lagrangian cycles on orbifolds, arXiv:1110.3866.
  • [16] R. P. Thomas, A holomorphic Casson invariant for Calabi–Yau 3-folds, and bundles on K3 fibrations, J. Differential Geom. 54 (2000), 367–438.