Commit 04dfa4fa authored by Jannes Bantje's avatar Jannes Bantje

start work on complex genera

parent 6cc928f0
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......@@ -288,4 +288,24 @@ This seems to be mostly algebra and I'm unsure, whether it is actually relevant
\todo{looked at chapter 3 in the book}
\todo[inline]{maybe not as important for my talk!?}
% section formal_group_laws_leon (end)
\section{Index Theory -- Jannes} % (fold)
\label{sec:index_theory}
Before we can get started, we need yet another genus to get things going.
In order to define this properly we need to talk about \emph{complex} genera: in \cref{sec:introduction_julian} we used the real cobordism ring $\Omega^{\SO}_*$ as the domain for genera.
Defining an analogue for complex manifolds is a little more complicated, since a cobordism between complex manifolds cannot admit a complex structure for dimension reasons.
This can however be done for \Index[stably almost complex manifold]{stably almost complex manifolds}.
\Textcite{novikov_complex_bordism,milnor_complex_bordism} proved results similar to the ones in \cref{sec:introduction_julian} for the complex bordism ring.\footnote{see \url{http://www.map.mpim-bonn.mpg.de/Complex_bordism} for a nice overview}
It turns out, that there is an analogous one-to-one correspondence between complex genera and power series $Q(x)$, but these need no longer be even!
\begin{example}
We consider the formal power series
\[
Q(x) = \frac{x}{1- e^{-x}} = 1 + \frac{1}{2} x + \frac{1}{12} x^2 + \ldots
\]
\end{example}
% section index_theory (end)
% chapter elliptic_genera_phd_seminar (end)
\ No newline at end of file
......@@ -922,4 +922,22 @@
year = {1958},
pages = {137--154},
url = {http://www.numdam.org/item?id=BSMF_1958__86__137_0},
}
@article{novikov_complex_bordism,
author = {Novikov, S. P.},
title = {Homotopy properties of {T}hom complexes},
journal = {Mat. Sb. (N.S.)},
volume = {57 (99)},
year = {1962},
pages = {407--442}
}
@article{milnor_complex_bordism,
author = {Milnor, J.},
title = {On the cobordism ring {$\Omega^{\ast}$} and a complex analogue. {I}},
journal = {American Journal of Mathematics},
volume = {82},
year = {1960},
pages = {505--521},
doi = {10.2307/2372970},
}
\ No newline at end of file
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