A law of conservation of number for local Euler characteristics

Abstract

Let T and V be complex analytic spaces, with T reduced and locally irreducible and π:T×V→T the projection map to the first factor. Let K ∗ be a complex 0→K n → X n K n−1 → X n−1 …→ X 2 K 1 → X 1 K 0 →0 of O T×V coherent sheaves, where all X j are O T×V -linear, all sheaves K j are O T -flat and such that the support of the homology sheaves H j (K ∗ ) is π -finite. For every t∈T , V t denotes {t}×V≅V ; K ∗ t denotes the complex obtained by tensoring K ∗ with O V t ; K ∗ t;p denotes the complex formed by the germs at (t,p) of K ∗ t and H j (K ∗ t;p ) denotes the j th homology group of the complex K ∗ t;p . The Euler characteristic of the complex of sheaves K ∗ t at a point (t,p)∈V t is defined as χ(K ∗ t;p )=∑ j=0 n (−1) j dim C H j (K ∗ t;p ). The authors show that, for every (t 0 ,p 0 )∈T×V there are neighbourhoods T ′ and V ′ of t 0 and p 0 , respectively, such that for every t∈T ′ , χ(K ∗ t 0 ;p 0 )=∑ q∈V ′ χ(K ∗ t;p ) .