# Algebraic Geometry 3 Curves Jaobians by Parshin Shafarevich

By Parshin Shafarevich

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Example text

Namely we have to show that for ξ, ψ ∈ V ∞ (X) one has µ(ξ, ψ) = ξ · ψ where the last product is understood in the usual sense. Let φ ∈ Vc∞ (X). Then we have < µ(ξ, ψ), φ >=< ψ, ξ · φ >= ψ · (ξ · φ) = (ξ · ψ) · φ =< ξ · ψ, φ > . Hence µ(ξ, φ) = ξ · ψ. Since V ∞ (X) is dense in V −∞ (X) and µ is continuous it follows that µ deﬁnes V ∞ (X)-module structure on V −∞ (X). 2 The Sheaf Property of Generalized Valuations In this subsection we describe the canonical sheaf structure on generalized valuations.

N F(X) ∩ Wi V −∞ (X) = Wi F(X) . Theory of Valuations on Manifolds, IV 41 Proof. (i) By the Hahn–Banach theorem it is enough to prove that for any φ ∈ Vc∞ (X)\{0} there exists f ∈ F(X) such that < f, φ >= 0. Let us ﬁx φ ∈ Vc∞ (X)\{0}. One may ﬁnd an open subset U ⊂ X and a real analytic diﬀeomorphism g : U →R ˜ n such that φ|U ≡ 0. The smooth valuation g∗ φ|U ∈ V ∞ (Rn ) does not vanish identically. 10 from [A5] there exists a convex compact set K ∈ K(Rn )∩P(Rn ) such that (g∗ φ)(K) = 0. Since every compact set can be approximated in the Hausdorﬀ metric by convex compact polytopes, we may assume that K is a convex compact polytope, and hence a subanalytic set.

N, and any f ∈ Wi (F(X))\ Wi+1 (F(X)) there exists φ ∈ Wn−i (Vc∞ (X)) such that < Ξ(f ), φ >= 0. Proof. Clearly it is enough to prove the second statement. Let us ﬁx a constructible function f ∈ Wi (F(X))\Wi+1 (F(X)). Thus supp f is a subanalytic set and codim(supp f ) = i. One can choose a regular point x ∈ supp f , a neighborhood U , a real analytic diﬀeomorphism g : U →R ˜ n such that f |U ◦ g −1 = c · 1lRn−k where Rn−k ⊂ Rn is the coordinate subspace, and c = 0 is a constant. Thus we may assume that X = Rn , f = 1lRn−k .