Grothendieck–Ogg–Shafarevich formula

In mathematics, the Grothendieck–Ogg–Shafarevich formula describes the Euler characteristic of a complete curve with coefficients in an abelian variety or constructible sheaf, in terms of local data involving the Swan conductor. Andrew Ogg (1962) and Igor Shafarevich (1961) proved the formula for abelian varieties with tame ramification over curves, and Alexander Grothendieck (1977, Exp. X formula 7.2) extended the formula to constructible sheaves over a curve (Raynaud 1965).

Statement

Suppose that F is a constructible sheaf over a genus g smooth projective curve C, of rank n outside a finite set X of points where it has stalk 0. Then

[math]\displaystyle{ \chi(C,F) = n(2-2g) -\sum_{x\in X}(n+Sw_x(F)) }[/math]

where Sw is the Swan conductor at a point.

References

• Grothendieck, Alexandre (1977) (in fr), Séminaire de Géométrie Algébrique du Bois Marie – 1965–66 – Cohomologie l-adique et Fonctions L – (SGA 5), Lecture notes in mathematics, 589, Berlin; New York: Springer-Verlag, pp. xii+484, doi:10.1007/BFb0096802, ISBN 3540082484 
• Ogg, Andrew P. (1962), "Cohomology of abelian varieties over function fields", Annals of Mathematics, Second Series 76: 185–212, doi:10.2307/1970272, ISSN 0003-486X 
• Raynaud, Michel (1965), "Caractéristique d'Euler–Poincaré d'un faisceau et cohomologie des variétés abéliennes", Séminaire Bourbaki, Vol. 9, Exp. No. 286, Paris: Société Mathématique de France, pp. 129–147, http://www.numdam.org/item?id=SB_1964-1966__9__129_0 
• Shafarevich, Igor R. (1961), "Principal homogeneous spaces defined over a function field", Akademiya Nauk SSSR. Trudy Matematicheskogo Instituta imeni V. A. Steklova 64: 316–346, ISSN 0371-9685