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
χ
(
C
,
F
)
=
n
(
2
−
2
g
)
−
∑
x
∈
X
(
n
+
S
w
x
(
F
)
)
{\displaystyle \chi (C,F)=n(2-2g)-\sum _{x\in X}(n+Sw_{x}(F))}
where Sw is the Swan conductor at a point.
References
Grothendieck, Alexandre (1977), Séminaire de Géométrie Algébrique du Bois Marie – 1965–66 – Cohomologie l-adique et Fonctions L – (SGA 5) , Lecture notes in mathematics (in French), vol. 589, Berlin; New York: Springer-Verlag , xii+484, doi :10.1007/BFb0096802 , ISBN 3540082484 Ogg, Andrew P. (1962), "Cohomology of abelian varieties over function fields", Annals of Mathematics 76 : 185– 212, doi :10.2307/1970272 , ISSN 0003-486X , JSTOR 1970272 , MR 0155824 Raynaud, Michel (1965), "Caractéristique d'Euler–Poincaré d'un faisceau et cohomologie des variétés abéliennes", Séminaire Bourbaki, Vol. 9 Société Mathématique de France , pp. 129– 147, MR 1608794 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 , MR 0162806 