Commit b0dfbd3d authored by Jannes Bantje's avatar Jannes Bantje

add example of L-genus

parent 29f4d18a
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......@@ -132,7 +132,7 @@ This can be used to define a genus:
One obviously has to check, that this is in fact a genus, see \cite[14]{hirzebruch_modularforms}.
One can now define $f(x) \coloneqq x/Q(x)$ and obtains an odd power series with coefficients in $R$.
One can now define $f(x) \coloneqq x/Q(x)$ and by doing so obtain an odd power series with coefficients in $R$.
Let $g$ be the (formal) inverse of $f$, called the \Index{logarithm of the genus $\varphi_Q$}.
By resorting to the total Chern and Pontryagin classes of $\mathbb{C}\mathbb{P}^n$ in \cref{thm:axioms_chern,prop:axioms-pontryagin} and a tricky computation, one can show, that
\[
......@@ -149,6 +149,16 @@ We therefore have
\[
\varphi(M) = \varphi(\Tang M)[M]
\]
\begin{example}
Letting $Q(x)=x/\tanh(x)$ yields an even power series (look at the Taylor expansion).
Following our earlier definitions, we get $f(x)=\tanh(x)$, $f'(x)=1-f(x)^2$ and hence
\[
g'(y)= \frac{1}{1-y^2} = 1+ y^2 +y^4 + \ldots
\]
The genus obtained by this power series is called \Index[L-genus@$L$-genus]{$L$-genus}.
It was proven by \textcite{hirzebruch_Lgenus}, that this genus coincides with the signature.
\end{example}
% section introduction_by_julian (end)
\section{Ellipctic Genera -- Markus} % (fold)
......
......@@ -905,6 +905,15 @@
doi = {10.1007/978-3-663-14045-0},
}
@book{hirzebruch_Lgenus,
author = {Hirzebruch, F.},
title = {Neue topologische {M}ethoden in der algebraischen {G}eometrie},
series = {Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Heft 9},
publisher = {Springer},
year = {1956},
pages = {viii+165},
}
@article{grothendieck_chern,
author = {Grothendieck, Alexander},
title = {La théorie des classes de {C}hern},
......
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