Virtual fundamental classes

One basic component of Gromov-Witten theory is the virtual fundamental class associated to a perfect obstruction theory. We will present some of the history and recent developments in this area.

The seminar will take place online via Zoom; we meet on Tuesdays, 16-18 Uhr. If you are interested in attending the seminar or giving one of the lectures, please contact me (Marc Levine) at

Previous Motives Seminars

Follow the Teaching link at the top of this page to find links to previous motives seminars.


Lecture 1 (Nov. 3-Marc Levine). Motivation and background from Gromov-Witten theory
Lecture 1 Slides
Lecture 1 Video

Lecture 2 (Nov. 10-Fangzhou Jin). Chern-Schwartz-MacPherson classes I [M], [GS], [Gi]
Lecture 2 Slides
Lecture 2 Video

Lecture 2-1/2(Nov. 17-Fangzhou Jin). Chern-Schwartz-MacPherson classes I.5 [M], [GS], [Gi]
Lecture 2-1/2 Slides

Lecture 3. (Nov. 17-Dhyan Aranha) A very brief bit of background on stacks and derived algebraic geometry. [T] ]
Lecture 3 Notes
Lecture 3 A correction
Video for Lectures 2-1/2 and 3.

Lecture 4. (Nov. 24-Charanya Ravi) The Behrend-Fantechi virtual fundamental class [BF]
Lecture 4 Notes
Video for Lecture 4.

Lecture 5 (Dec. 1-Maria Yakerson) Localization of virtual classes [GP]
Lecture 5 Notes
Video for Lecture 5.

Lectures 6/7. (Dec. 8, 15-Ran Azouri) Behrend's work on symmetric obstruction theories [B], [BBS]
Lecture 6 Notes
Video for Lecture 6.
Lecture 7 Notes
Video for Lecture 7.

Lecture 8. (Dec. 22-Viktor Tabakov) Deglise-Jin-Khan Fundamental classes [DJK]
Lecture 8 Slides
Video for Lecture 8.

Lecture 9. (Jan. 12-Federico Binda) Virtual classes for Artin stacks [AP]
Lecture 9 Notes
Video for Lecture 9.

Lecture 10. (Jan. 19-Dhyan Aranha) Khan's virtual classes for quasi-smooth morphisms [K]
Lecture 10 Notes
Video for Lecture 10.

Lecture 11. (Jan. 26-Alessandro D'Angelo) A comparison of the virtual fundamental classes of Khan and of Behrend-Fantechi
Lecture 11 Notes
Video for Lecture 11.

Lecture 12. (Feb. 2-Sabrina Pauli) Virtual fundamental classes in motivic homotopy theory [L]
Lecture 12 Notes
Video for Lecture 12.


[AP] Dhyan Aranha, Piotr Pstragowski, The Intrinsic Normal Cone For Artin Stacks arXiv:1909.07478 [math.AG]

[B] Behrend, Kai, Donaldson-Thomas type invariants via microlocal geometry. Ann. of Math. (2) 170 (2009), no. 3, 1307-1338.

[BBS] Behrend, Kai; Bryan, Jim; Szendröi, Balázs, Motivic degree zero Donaldson-Thomas invariants. Invent. Math. 192 (2013), no. 1, 111-160.

[BF] Behrend, K.; Fantechi, B. The intrinsic normal cone. Invent. Math. 128 (1997), no. 1, 45-88.

[DJK] Frédéric Déglise, Fangzhou Jin, Adeel A. Khan, Fundamental classes in motivic homotopy theory arXiv:1805.05920 [math.AG math.KT]

[Gi] V. Ginsburg, Characteristic varieties and vanishing cycles. Invent. Math. 84 (1986), no. 2, 327–402.

[GS] González-Sprinberg, Gerardo, L'obstruction locale d'Euler et le théorème de MacPherson, pp. 7-32,

[GP] T. Graber, R. Pandharipande, Localization of virtual classes Inventiones mathematicae volume 135, pages 487-518(1999)

[K] Adeel A. Khan, Virtual fundamental classes of derived stacks I arXiv:1909.01332 [math.AG]

[L] Marc Levine, The intrinsic stable normal cone arXiv:1703.03056 [math.AG]

[M] MacPherson, R. D. Chern classes for singular algebraic varieties. Ann. of Math. (2) 100 (1974), 423-432.

Astérisque, 83–83, Soc. Math. France, Paris, 1981.

[T] B. Toen, Derived algebraic geometry