Grothendieck–Ogg–Shafarevich formula

Template:Distinguish 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. Template:Harvs and Template:Harvs proved the formula for abelian varieties with tame ramification over curves, and Template:Harvs extended the formula to constructible sheaves over a curve Template:Harv.

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

$\chi (C,F)=n(2-2g)-\sum _{x\in X}(n+S{w}_x(F))$

where Sw is the Swan conductor at a point.

References

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