Bivariant class of degree one

TL;DR

This paper studies bivariant classes of degree one in the context of projective birational morphisms, establishing formulas relating the (co)homology of the involved varieties and characterizing when the target is an A-homology manifold.

Contribution

It introduces the concept of degree one bivariant classes, derives explicit formulas extending classic blowing-up formulas, and characterizes when the target variety is an A-homology manifold based on the existence of such classes.

Findings

Explicit formulas relating (co)homology of X and Y

Characterization of Y as an A-homology manifold via degree one classes

Uniqueness of the degree one bivariant class when Y is an A-homology manifold

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 $f^{-1}({\text{Sing}}(Y))$.