Bivariant Class of Degree One

Let f : X → Y be a projective birational morphism, between complex quasi-projective varieties. Fix a bivariant class θ ∈ H0(X →f Y ) ∼= H o m D cb ( Y ) ( R f ∗ A X , A Y ) ( h e r e A i s a N o e t h e r i a n c o m m u t a t i v e r i n g with identity, and AX and AY denote the constant sheaves). Let θ0 : H0(X) → H0(Y) be the induced Gysin morphism. We say that θhas degree one if θ0(1X) = 1Y ∈ H0(Y). This is equivalent to say that θ is a section of the pull-back f∗ : AY → Rf∗AX, i.e. θ◦f∗ = idAY , and it is also equivalent to say that AY is a direct summand of Rf∗AX. We investigate the consequences of the existence of a bivariant class of degree one. We prove explicit formulas relating the (co)homology of X and Y , which extend the classic formulas of the blowing-up. These formulas are compatible with the duality morphism. Using which, we prove that the existence of a bivariant class θ of degree one for a resolution of singular- ities, is equivalent to require that Y is an A-homology manifold. In this case θ is unique, and the Betti numbers of the singular locus Sing(Y ) of Y are related with the ones of f−1(Sing(Y )).