Chern classes as bivariant classes

Easier than in the case of schemes

spaces-chow.tex

@@ -4363,6 +4363,179 @@ \section{Projective space bundle formula}
 
 
 
+\section{The Chern classes of a vector bundle}
+\label{section-chern-classes-vector-bundles}
+
+\noindent
+This section is the analogue of Chow Homology, Sections
+\ref{chow-section-chern-classes-vector-bundles} and
+\ref{chow-section-intersecting-chern-classes}.
+However, contrary to what is done there, we directly
+define the chern classes of a vector bundle as bivariant classes.
+This saves a considerable amount of work.
+
+\begin{lemma}
+\label{lemma-segre-classes}
+In Situation \ref{situation-setup} let $X/B$ be good.
+Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$.
+Let $(\pi : P \to X, \mathcal{O}_P(1))$ be the projective space
+bundle associated to $\mathcal{E}$. For every
+morphism $X' \to X$ of good algebraic spaces over $B$
+there are unique maps
+$$
+c_i(\mathcal{E}) \cap - : A_k(X') \longrightarrow A_{k - i}(X'),\quad
+i = 0, \ldots, r
+$$
+such that for $\alpha \in A_k(X')$ we have
+$c_0(\mathcal{E}) \cap \alpha = \alpha$ and
+$$
+\sum\nolimits_{i = 0, \ldots, r}
+(-1)^i c_1(\mathcal{O}_{P'}(1))^i \cap
+(\pi')^*\left(c_{r - i}(\mathcal{E}) \cap \alpha\right) = 0
+$$
+where $\pi' : P' \to X'$ is the base change of $\pi$.
+Moreover, these maps define a bivariant class
+$c_i(\mathcal{E})$ of degree $i$ on $X$.
+\end{lemma}
+
+\begin{proof}
+Uniqueness and existence of the maps $c_i(\mathcal{E}) \cap -$
+follows immediately from Lemma \ref{lemma-chow-ring-projective-bundle}
+and the given description of $c_0(\mathcal{E})$. For every $i \in \mathbf{Z}$
+the rule which to every morphism $X' \to X$ of good algebraic spaces
+over $B$ assigns the map
+$$
+t_i(\mathcal{E}) \cap - :
+A_k(X') \longrightarrow A_{k - i}(X'),\quad
+\alpha \longmapsto
+\pi'_*(c_1(\mathcal{O}_{P'}(1))^{r - 1 + i} \cap (\pi')^*\alpha)
+$$
+is a bivariant class\footnote{Up to signs these are
+the Segre classes of $\mathcal{E}$.} by Lemmas \ref{lemma-cap-c1-bivariant},
+\ref{lemma-flat-pullback-bivariant}, and
+\ref{lemma-push-proper-bivariant}.
+By Lemma \ref{lemma-cap-projective-bundle} we have
+$t_i(\mathcal{E}) = 0$ for $i < 0$ and $t_0(\mathcal{E}) = 1$.
+Applying pushforward to the equation in the statement of the lemma
+we find from Lemma \ref{lemma-cap-projective-bundle} that
+$$
+(-1)^r t_1(\mathcal{E}) + (-1)^{r - 1}c_1(\mathcal{E}) = 0
+$$
+In particular we find that $c_1(\mathcal{E})$ is a bivariant class.
+If we multiply the equation in the statement of the lemma by
+$c_1(\mathcal{O}_{P'}(1))$ and push the result forward to $X'$
+we find
+$$
+(-1)^r t_2(\mathcal{E}) +
+(-1)^{r - 1} t_1(\mathcal{E}) \cap c_1(\mathcal{E}) +
+(-1)^{r - 2} c_2(\mathcal{E}) = 0
+$$
+As before we conclude that $c_2(\mathcal{E})$ is a bivariant class.
+And so on.
+\end{proof}
+
+\begin{definition}
+\label{definition-chern-classes}
+In Situation \ref{situation-setup} let $X/B$ be good.
+Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$.
+For $i = 0, \ldots, r$ the {\it $i$th chern class of $\mathcal{E}$}
+is the bivariant class $c_i(\mathcal{E}) \in A^i(X)$ of degree $i$
+constructed in Lemma \ref{lemma-segre-classes}.
+\end{definition}
+
+\noindent
+For convenience we often set $c_i(\mathcal{E}) = 0$
+for $i > r$ and $i < 0$. By definition
+we have $c_0(\mathcal{E}) = 1 \in A^0(X)$.
+Here is a sanity check.
+
+\begin{lemma}
+\label{lemma-first-chern-class}
+In Situation \ref{situation-setup} let $X/B$ be good.
+Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module.
+The first chern class of $\mathcal{L}$ on $X$ of
+Definition \ref{definition-chern-classes}
+is equal to the bivariant class of Lemma \ref{lemma-cap-c1-bivariant}.
+\end{lemma}
+
+\begin{proof}
+Namely, in this case $P = \mathbf{P}(\mathcal{L}) = X$ with
+$\mathcal{O}_P(1) = \mathcal{L}$ by our normalization of the
+projective bundle, see Section \ref{section-projective-space-bundle-formula}.
+Hence the equation in Lemma \ref{lemma-segre-classes}
+reads
+$$
+(-1)^0 c_1(\mathcal{L})^0 \cap c^{new}_1(\mathcal{L}) \cap \alpha +
+(-1)^1 c_1(\mathcal{L})^1 \cap c^{new}_0(\mathcal{L}) \cap \alpha = 0
+$$
+where $c_i^{new}(\mathcal{L})$ is as in
+Definition \ref{definition-chern-classes}.
+Since $c_0^{new}(\mathcal{L}) = 1$ and $c_1(\mathcal{L})^0 = 1$
+we conclude.
+\end{proof}
+
+\noindent
+Next we see that chern classes are in the center of the bivariant
+Chow cohomology ring $A^*(X)$.
+
+\begin{lemma}
+\label{lemma-cap-commutative-chern}
+In Situation \ref{situation-setup} let $X/B$ be good.
+Let $\mathcal{E}$ be a locally free $\mathcal{O}_X$-module of rank $r$.
+Then $c_j(\mathcal{L}) \in A^j(X)$ commutes with every
+element $c \in A^p(X)$. In particular, if $\mathcal{F}$ is a
+second locally free $\mathcal{O}_X$-module on $X$ of rank $s$, then
+$$
+c_i(\mathcal{E}) \cap c_j(\mathcal{F}) \cap \alpha
+=
+c_j(\mathcal{F}) \cap c_i(\mathcal{E}) \cap \alpha
+$$
+as elements of $A_{k - i - j}(X)$ for all $\alpha \in A_k(X)$.
+\end{lemma}
+
+\begin{proof}
+Let $X' \to X$ be a morphism of good algebraic spaces over $B$.
+Let $\alpha \in A_k(X')$. Write $\alpha_j = c_j(\mathcal{E}) \cap \alpha$, so
+$\alpha_0 = \alpha$. By Lemma \ref{lemma-segre-classes} we have
+$$
+\sum\nolimits_{i = 0}^r
+(-1)^i c_1(\mathcal{O}_{P'}(1))^i \cap
+(\pi')^*(\alpha_{r - i}) = 0
+$$
+in the chow group of the projective bundle
+$(\pi' : P' \to X', \mathcal{O}_{P'}(1))$
+associated to $(X' \to X)^*\mathcal{E}$.
+Applying $c \cap -$ and using Lemma \ref{lemma-c1-center}
+and the properties of bivariant classes we obtain
+$$
+\sum\nolimits_{i = 0}^r
+(-1)^i c_1(\mathcal{O}_{P'}(1))^i \cap
+\pi^*(c \cap \alpha_{r - i}) = 0
+$$
+in the Chow group of $P'$. Hence we see that $c \cap \alpha_j$ is
+equal to $c_j(\mathcal{E}) \cap (c \cap \alpha)$ by the uniqueness in
+Lemma \ref{lemma-segre-classes}. This proves the lemma.
+\end{proof}
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 \input{chapters}