Bivariant Class of Degree One

Abstract

Let f:Xrightarrow 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) (here {mathbb {A}} is a Noetherian commutative ring with identity, and {mathbb {A}}_X and {mathbb {A}}_Y denote the constant sheaves). Let theta _0:H^0(X)rightarrow H^0(Y) be the induced Gysin morphism. We say that theta has degree one if theta _0(1_X)= 1_Yin H^0(Y). This is equivalent to say that theta is a section of the pull-back f^*: {mathbb {A}}_Yrightarrow Rf_*{mathbb {A}}_X, i.e. theta circ f^*={text {id}}_{{mathbb {A}}_Y}, and it is also equivalent to say that {mathbb {A}}_Y is a direct summand of Rf_*{mathbb {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 of degree one for a resolution of singularities, is equivalent to require that Y is an {mathbb {A}}-homology manifold. In this case theta is unique, and the Betti numbers of the singular locus {text {Sing}}(Y) of Y are related with the ones of f^{-1}({text {Sing}}(Y)).