## 81.28 The Chern classes of a vector bundle

This section is the analogue of Chow Homology, Sections 42.37 and 42.38. 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.

Lemma 81.28.1. In Situation 81.2.1 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

$c_ i(\mathcal{E}) \cap - : \mathop{\mathrm{CH}}\nolimits _ k(X') \longrightarrow \mathop{\mathrm{CH}}\nolimits _{k - i}(X'),\quad i = 0, \ldots , r$

such that for $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X')$ 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 $X$.

Proof. Uniqueness and existence of the maps $c_ i(\mathcal{E}) \cap -$ follows immediately from Lemma 81.27.2 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

$t_ i(\mathcal{E}) \cap - : \mathop{\mathrm{CH}}\nolimits _ k(X') \longrightarrow \mathop{\mathrm{CH}}\nolimits _{k - i}(X'),\quad \alpha \longmapsto \pi '_*(c_1(\mathcal{O}_{P'}(1))^{r - 1 + i} \cap (\pi ')^*\alpha )$

is a bivariant class1 by Lemmas 81.26.4, 81.26.5, and 81.26.7. By Lemma 81.27.1 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 81.27.1 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. $\square$

Definition 81.28.2. In Situation 81.2.1 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 $i$th Chern class of $\mathcal{E}$ is the bivariant class $c_ i(\mathcal{E}) \in A^ i(X)$ of degree $i$ constructed in Lemma 81.28.1. The total Chern class of $\mathcal{E}$ is the formal sum

$c(\mathcal{E}) = c_0(\mathcal{E}) + c_1(\mathcal{E}) + \ldots + c_ r(\mathcal{E})$

which is viewed as a nonhomogeneous bivariant class on $X$.

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.

Lemma 81.28.3. In Situation 81.2.1 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 81.28.2 is equal to the bivariant class of Lemma 81.26.4.

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 81.27. Hence the equation in Lemma 81.28.1 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 81.28.2. Since $c_0^{new}(\mathcal{L}) = 1$ and $c_1(\mathcal{L})^0 = 1$ we conclude. $\square$

Next we see that Chern classes are in the center of the bivariant Chow cohomology ring $A^*(X)$.

Lemma 81.28.4. In Situation 81.2.1 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 $\mathop{\mathrm{CH}}\nolimits _{k - i - j}(X)$ for all $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$.

Proof. Let $X' \to X$ be a morphism of good algebraic spaces over $B$. Let $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X')$. Write $\alpha _ j = c_ j(\mathcal{E}) \cap \alpha$, so $\alpha _0 = \alpha$. By Lemma 81.28.1 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 81.26.8 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 81.28.1. This proves the lemma. $\square$

Remark 81.28.5. In Situation 81.2.1 let $X/B$ be good. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module. If the rank of $\mathcal{E}$ is not constant then we can still define the Chern classes of $\mathcal{E}$. Namely, in this case we can write

$X = X_0 \amalg X_1 \amalg X_2 \amalg \ldots$

where $X_ r \subset X$ is the open and closed subspace where the rank of $\mathcal{E}$ is $r$. If $X' \to X$ is a morphism of good algebraic spaces over $B$, then we obtain by pullback a corresponding decomposition of $X'$ and we find that

$\mathop{\mathrm{CH}}\nolimits _*(X') = \prod \nolimits _{r \geq 0} \mathop{\mathrm{CH}}\nolimits _*(X'_ r)$

by our definitions. Then we simply define $c_ i(\mathcal{E})$ to be the bivariant class which preserves these direct product decompositions and acts by the already defined operations $c_ i(\mathcal{E}|_{X_ r}) \cap -$ on the factors. Observe that in this setting it may happen that $c_ i(\mathcal{E})$ is nonzero for infinitely many $i$.

 Up to signs these are the Segre classes of $\mathcal{E}$.

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