Commit cc1644bc authored by Jannes Bantje's avatar Jannes Bantje

introduce Chern character properly

parent 82d9a2da
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......@@ -82,7 +82,7 @@ The general idea is to consider Chern resp. Pontryagin classes formally as eleme
\[
c(E) = (1 + x_1) \cdots (1+ x_n) = 1 + c_1 + \ldots + c_r
\]
where the Chern class $c_r= \sigma(x_1, \ldots ,x_n)$ is the $r$-th \Index{elementary symmetric function} in the $x_i$.
where the Chern class $c_r= \sigma_r(x_1, \ldots ,x_n)$ is the $r$-th \Index{elementary symmetric function} in the $x_i$.
By assuming the existence of Hermitian structures on the $E_1$, we can reduce their structure groups from $\mathbb{C}^\times$ to $T^1=S^1\cong\Unitary(1)$ and therefore the one of $E$ to
\[
T^n = \set*{A \in \Unitary(n) \given A = \operatorname{diag} \enbrace*{e^{2 \pi i \varphi_1},\ldots , e^{2 \pi i \varphi_n}}, \varphi_i \in \mathbb{R}}
......@@ -326,7 +326,47 @@ From a computational point of view, one could say, that we now plug the Chern cl
References: \cite[Ex.~III.11.11]{lawson_spin}
\end{example}
The next player, that we need too, is the \Index{Chern character}.
The next player, that we need too, is the \Index{Chern character}:
Let $E$ be a complex vector bundle of dimension $n$ over $X$.
We may write the total rational Chern class formally as\marginnote{$\deg x_i=2$}
\[
c(E) = 1 + c_1 + \ldots + c_n = (1+x_1) \cdots (1+x_n)
\]
so that $c_i = \sigma_i(x_1, \ldots ,x_n)$ by the splitting principle.
\begin{definition}
The expression
\[
\ch(E) = e^{x_1} + \ldots + e^{x_n} = n + \sum_{j=1}^{n} x_j + \frac{1}{2} \sum_{j=1}^{n} x_j^2 + \ldots
\]
is called the \Index{Chern character} of $E$.
\end{definition}
The term of degree $k$ in $\ch(E)$ is the symmetric polynomial $\ch^kE = \frac{1}{k!} \sum_{j=1}^{n} x^k_j$, which therefore can be written as a polynomial in the elementary symmetric functions $c_1, \ldots ,c_n$.
In particular
\[
\ch E = n + \ch^1 E + \ch^2 E + \ldots \in H^{2^*}(X;\mathbb{Q})
\]
is well-defined.
Note, that for a complex line bundle $L$ we have $\ch(L) = e^{c_1(L)}$.
The nice thing about the Chern character is the following:
\begin{proposition}
For complex vector bundles $E,E'$ over $X$, we have
\begin{enumerate}[(i)]
\item $\ch(E \oplus E') = \ch(E) + \ch(E')$
\item $\ch(E \otimes E') = \ch(E) \ch(E')$
\end{enumerate}
\end{proposition}
This gives a ring homomorphism
\[
\ch \colon \K(X) \To{} H^{2^*}(X;\mathbb{Q})
\]
\begin{remark}
This can be generalised \emph{drastically} to generalised cohomology theories! This is the arena of the Atiyah--Hirzebruch spectral sequence.
\end{remark}
% section index_theory (end)
% chapter elliptic_genera_phd_seminar (end)
\ No newline at end of file
......@@ -232,6 +232,7 @@
% \newcommand{\M}{\mathrm{M}\mkern-3mu}
\newcommand{\thoM}{M\mkern-3mu}
\DeclareMathOperator{\td}{td}
\DeclareMathOperator{\ch}{ch}
\newcommand{\tmf}{\ensuremath{\mathrm{tmf}}}
\DeclareMathOperator{\Map}{Map}
\newcommand{\sa}{\mathrm{sa}}
......
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