 ... ... @@ -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{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) ... ...