Seminar on Bloch's conductor formula

Tuesdays, WSC-S-U-4.02, 14-16 Uhr

In his papers [B1, B2] Bloch proposed a formula for the difference between the étale Euler characteristic of the general fiber and special fiber of a generically smooth proper morphism of a regular scheme $X$ over a DVR $\mathcal{O}$ with quotient field $K$ in terms of a localised self-intersection number of the diagonal on $X\times_{\mathcal{O}}X$ and the so-called Swan conductor of the generic fiber over $K$. Bloch proved his formula in the case of curves. Under additional hypothesis, Bloch's formula was proven by K. Kato and T. Saito in [KS]. Besides the intrinsic interest of Bloch's formula, this subject has a great deal to do with wild ramification phenomena and its correspondence with differential equations with irregular singularities in characteristic 0.

In this seminar, we plan to first cover some introductory material on ramification theory, the Artin conductor and the Swan conductor, as well other classical related topics such as the formula of Grothendieck-Ogg-Shafarevich. We will discuss Bloch's papers [B1, B2] and the paper [KS] of Kato and Saito. If time remains, we will also discuss the paper of Ogogorzo [O] or get an introduction to the theory of characteristic cycles and singular support from the paper of Saito [S].

There is a "non-commutative'' approach to this subject, detailed in the paper [TV] of Toën and Vezzosi. We will not have time to get into this paper, but may carry over the seminar to the next semester with this as a main theme.

Preliminary Program of Lectures

1. 15.10 (Marc Levine) Overview lecture

2. 22.10 (Tariq Syed) Some background: Ramification theory, Artin representation, Artin conductor, Swan conductor. [Serre 1, 2]

3. 29.10 (Viktor Kleen) Euler characteristic of a torsion sheaf on a curve. [Raynaud]

4. 05.11 (Fangzhou Jin) The formula of Ogg-Shafarevich-Grothendieck. [Raynaud].

5. 12.11 (Daniel Harrer) Bloch's conductor formula 1 [B1]

6. 19.11 (Gabriela Guzman) Bloch's condutor formula 2 [B1]

7. 26.11 (Enzo Serandon) Cotangent complexes and the Atiyah classes [KS, sect 1.1-1.4].

8. 03.12 (Pavel Sechin) Properties of the Atiyah classes [KS, sect. 1.4-1.7]

9. 10.12 (Alessandro D'Angelo) K-theory and localized Chern classes [KS, sect. 2].

10 17.12 (Heng Xie) K-theoretic localised intersection product [KS, sect. 3]

11. 07.01.2020 (Heer Zhao) Logarithmic products [KS, sect. 4]

12. 14.01 Localised intersection product [KS, sect. 5]

13. 21.01 (Manh Toan Nguyen) Conductor formula [KS, sect. 6]

14. 28.01 (Ran Azouri) Bloch's conjecture and Milnor numbers [O].

References

[B1] Bloch, Spencer, Cycles on arithmetic schemes and Euler characteristics of curves. Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 421–450, Proc. Sympos. Pure Math., 46, Part 2, Amer. Math. Soc., Providence, RI, 1987.

[B2] Bloch, Spencer, Euler characteristics and Swan conductors. Algebraic geometry, Sendai, 1985, 85–90, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987.

[KS] Kato, Kazuya; Saito, Takeshi, On the conductor formula of Bloch. Publ. Math. Inst. Hautes Études Sci. No. 100 (2004), 5–151.

[O] Orgogozo, Fabrice, Conjecture de Bloch et nombres de Milnor. Ann. Inst. Fourier (Grenoble) 53 (2003), no. 6, 1739–1754.

[Raynaud] Michel Raynaud, Caractèristique d'Euler-Poincaré et cohomologie de varietétés abéliennes. Séminaire Bourbaki 17e année, 1964/65, no. 286

[S] Saito, Takeshi, The characteristic cycle and the singular support of a constructible sheaf. Invent. Math. 207 (2017), no. 2, 597–695.

[Serre 1] Serre, Jean-Pierre Corps locaux. Deuxième édition. Publications de l'Université de Nancago, No. VIII. Hermann, Paris, 1968. 245 pp.

[Serre 2] Serre, Jean-Pierre, Sur la rationalité des représentations d'Artin. Ann. of Math. (2) 72 (1960), 405–420.

[TV] B. Toën, G. Vezzosi, Géométrie non-commutativ, formule de trace et conducteur de Bloch, arXiv:1701.00455 [math.AG].