Bivariant class of degree one

Abstract

Let $f:X\to Y$ be a projective birational morphism, between complex quasi-projective varieties. Fix a bivariant class $\theta \in H^0(X\stackrel{f}\to Y)\cong Hom_{D^{b}_{c}(Y)}(Rf_*\mathbb A_X, \mathbb A_Y)$ (here $\mathbb A$ is a Noetherian commutative ring with identity, and $\mathbb A_X$ and $\mathbb A_Y$ denote the constant sheaves). Let $\theta_0:H^0(X)\to H^0(Y)$ be the induced Gysin morphism. We say that {\it $\theta$ has degree one} if $\theta_0(1_X)= 1_Y\in H^0(Y)$. This is equivalent to say that $\theta$ is a section of the pull-back $f^*: \mathbb A_Y\to Rf_*\mathbb A_X$, 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 $f^{-1}({\text{Sing}}(Y))$.