Example 42.42.6. For every $p \geq 1$ there is a unique homogeneous polynomial $P_ p \in \mathbf{Z}[c_1, c_2, c_3, \ldots ]$ of degree $p$ such that, for any $n \geq p$ we have

$P_ p(s_1, s_2, \ldots , s_ p) = \sum x_ i^ p$

in $\mathbf{Z}[x_1, \ldots , x_ n]$ where $s_1, \ldots , s_ p$ are the elementary symmetric polynomials in $x_1, \ldots , x_ n$, so

$s_ i = \sum \nolimits _{1 \leq j_1 < \ldots < j_ i \leq n} x_{j_1}x_{j_2} \ldots x_{j_ i}$

The existence of $P_ p$ comes from the well known fact that the elementary symmetric functions generate the ring of all symmetric functions over the integers. Another way to characterize $P_ p \in \mathbf{Z}[c_1, c_2, c_3, \ldots ]$ is that we have

$\log (1 + c_1 + c_2 + c_3 + \ldots ) = \sum \nolimits _{p \geq 1} (-1)^{p - 1}\frac{P_ p}{p}$

as formal power series. This is clear by writing $1 + c_1 + c_2 + \ldots = \prod (1 + x_ i)$ and applying the power series for the logarithm function. Expanding the left hand side we get

\begin{align*} & (c_1 + c_2 + \ldots ) - (1/2)(c_1 + c_2 + \ldots )^2 + (1/3)(c_1 + c_2 + \ldots )^3 - \ldots \\ & = c_1 + (c_2 - (1/2)c_1^2) + (c_3 - c_1c_2 + (1/3)c_1^3) + \ldots \end{align*}

In this way we find that

\begin{align*} P_1 & = c_1, \\ P_2 & = c_1^2 - 2c_2, \\ P_3 & = c_1^3 - 3c_1c_2 + 3c_3, \\ P_4 & = c_1^4 - 4c_1^2c_2 + 4c_1c_3 + 2c_2^2 - 4c_4, \end{align*}

and so on. Since the Chern classes of a finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$ are the elementary symmetric polynomials in the Chern roots $x_ i$, we see that

$P_ p(\mathcal{E}) = \sum x_ i^ p$

For convenience we set $P_0 = r$ in $\mathbf{Z}[r, c_1, c_2, c_3, \ldots ]$ so that $P_0(\mathcal{E}) = r(\mathcal{E})$ as a bivariant class (as in Remarks 42.37.10 and 42.42.5).

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0F9B. Beware of the difference between the letter 'O' and the digit '0'.