Bivariant Class of Degree One

Abstract

AbstractLet $$f:X\rightarrow Y$$ f : X → Y be a projective birational morphism, between complex quasi-projective varieties. Fix a bivariant class $$\theta \in H^0(X{\mathop {\rightarrow }\limits ^{f}}Y)\cong Hom_{D^{b}_{c}(Y)}(Rf_*{\mathbb {A}}_X, {\mathbb {A}}_Y)$$ θ ∈ H 0 ( X → f Y ) ≅ H o m D c b ( Y ) ( R f ∗ A X , A Y ) (here $${\mathbb {A}}$$ A is a Noetherian commutative ring with identity, and $${\mathbb {A}}_X$$ A X and $${\mathbb {A}}_Y$$ A Y denote the constant sheaves). Let $$\theta _0:H^0(X)\rightarrow H^0(Y)$$ θ 0 : H 0 ( X ) → H 0 ( Y ) be the induced Gysin morphism. We say that $$\theta $$ θ has degree one if $$\theta _0(1_X)= 1_Y\in H^0(Y)$$ θ 0 ( 1 X ) = 1 Y ∈ H 0 ( Y ) . This is equivalent to say that $$\theta $$ θ is a section of the pull-back $$f^*: {\mathbb {A}}_Y\rightarrow Rf_*{\mathbb {A}}_X$$ f ∗ : A Y → R f ∗ A X , i.e. $$\theta \circ f^*={\text {id}}_{{\mathbb {A}}_Y}$$ θ ∘ f ∗ = id A Y , and it is also equivalent to say that $${\mathbb {A}}_Y$$ A Y is a direct summand of $$Rf_*{\mathbb {A}}_X$$ R f ∗ A X . 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 $$\theta $$ θ of degree one for a resolution of singularities, is equivalent to require that Y is an $${\mathbb {A}}$$ A -homology manifold. In this case $$\theta $$ θ is unique, and the Betti numbers of the singular locus $${\text {Sing}}(Y)$$ Sing ( Y ) of Y are related with the ones of $$f^{-1}({\text {Sing}}(Y))$$ f - 1 ( Sing ( Y ) ) .