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


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