Commit fd76af4d authored by Jannes Bantje's avatar Jannes Bantje

further work

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%!TEX root = ../index_theory.tex
\chapter{Elliptic Genera -- Recapitulation from the Seminar} % (fold)
\chapter{Elliptic Genera -- Recapitulation of the Seminar} % (fold)
\label{cha:elliptic_genera_phd_seminar}
......@@ -96,7 +96,9 @@ The general idea is to consider Chern resp. Pontryagin classes formally as eleme
A similar story can be told for the Pontryagin classes, see \cite[8]{hirzebruch_modularforms}.
\end{example}
\Textcite[Sec.~1.5]{hirzebruch_modularforms} continue to define the \Index{Chern character} and prove general formulas, that can be applied to $\Lambda^k E$ or $E^*$ for example.
\todo[inline]{mention splitting principle}
% \Textcite[Sec.~1.5]{hirzebruch_modularforms} continue to define the \Index{Chern character} and prove general formulas, that can be applied to $\Lambda^k E$ or $E^*$ for example.
Let us systematically construct genera:
We consider an even power series
......@@ -140,7 +142,7 @@ By resorting to the total Chern and Pontryagin classes of $\mathbb{C}\mathbb{P}^
\]
that is, its derivative essentially determined by the values of $\varphi_Q$ at the complex projective spaces, see \cite[15]{hirzebruch_modularforms}.
But this also canonically associates a genus $\varphi$ with an even power series by looking at its values on $\CP^n$.
\begin{proposition}
\begin{proposition}\label{prop:one-to-one_genera}
There is a one-to-one correspondence of genera $\varphi$ and multiplicative sequences $Q$.
\end{proposition}
......@@ -151,7 +153,7 @@ We therefore have
\]
\begin{example}
Letting $Q(x)=x/\tanh(x)$ yields an even power series (look at the Taylor expansion).
Letting $Q(x)=\frac{x}{\tanh(x)}$ 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
......@@ -293,22 +295,33 @@ This seems to be mostly algebra and I'm unsure, whether it is actually relevant
\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}.
We start with a quick recap an genera as discussed in \cref{sec:introduction_julian}.
Following \textcite{hirzebruch_modularforms} a genus is a ring homomorphism
\[
\varphi \colon \Omega^{\SO}_* \otimes \mathbb{Q} \To{} R
\]
The main takeaway was, that even power series $Q(x) \in R\llbracket x\rrbracket$ (with constant term one) give rise to multiplicative sequences in which one can plug in the Pontryagin classes of an oriented, compact manifold to get a genus.
It even turned out, that there is a one-to-one correspondence of genera and such power series (see \cref{prop:one-to-one_genera}).
Before we can get started with index theory, we need to talk about the sibling of this correspondence.
Enter complex genera.
Cobordism between complex manifolds is somewhat more complicated\footnote{in particular a bordism between two complex manifolds cannot have a complex structure for dimension reasons}, but it works 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 $\Omega^{\Unitary}$.\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!
Nonetheless, on can proceed in the same fashion: let $X$ be a compact, almost complex manifold of real dimension $2n$.
Then we write
By applying the splitting principle we write the total Chern class as
\[
c(X) = 1 + c_1(X) + \ldots + c_n(X) = (1+ x_1) \cdots (1+ x_n)
\]
and get a genus out of $Q$ by setting $\varphi(X) = \enbrace[\big]{\prod_{i=1}^n Q(x_i)}[X]$.
From a computational point of view, one could say, that we now plug the Chern classes instead of the Pontryagin classes into a multiplicative sequence.
From a computational point of view, one could say, that we now just plug the Chern classes instead of the Pontryagin classes into a multiplicative sequence.
\textcite[Sec.~III.11]{lawson_spin} treat the real and complex case more side-by-side in comparison to \textcite{hirzebruch_modularforms}.
\todo[inline]{magic numbers $s_n$}
This digression into complex bordism arises from the need to define the following characteristic class, which features prominently in (classical) index theory.
\begin{example}
We consider the formal power series
......@@ -326,6 +339,23 @@ 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 Todd class can be viewed as the complex analogue of the \Ahat-class:
\begin{proposition}{\cite[Prop.~III.11.14]{lawson_spin}}
For any real oriented vector bundle $E$, the total Todd and \Ahat-classes satisfy
\[
\td(E \otimes \mathbb{C}) = \Ahat(E)^2
\]
\end{proposition}
\begin{proof}
This can be seen on the level of power series:
By the splitting principle we consider the formal splitting $E \otimes \mathbb{C} = \ell_1 \oplus \overline{\ell}_1 \oplus \ldots \oplus \ell_n \oplus \overline{\ell}_n$.
Then writing $x_j = c_1(\ell_j)$ we have
\[
\td(E \otimes \mathbb{C}) = \prod_{j=1}^n \frac{x_j}{1- e^{-x_j}} \frac{(-x_j)}{1- e^{x_j}} = \prod_{j=1}^n \enbrace*{\frac{x_j}{e^{x_j/2} - e^{-x_j/2}} }^2 = \prod_{j=1}^n \enbrace*{\frac{x_j/2}{\sinh(x_j/2)}}^2 = \Ahat(E)^2 \qedhere
\]
\end{proof}
The next player, that we need to state the index theorem, 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{$\deg x_i=2$}
......@@ -352,7 +382,7 @@ 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}
\begin{proposition}{\cite[Prop.~III.11.16]{lawson_spin}}
For complex vector bundles $E,E'$ over $X$, we have
\begin{enumerate}[(i)]
\item $\ch(E \oplus E') = \ch(E) + \ch(E')$
......@@ -366,7 +396,15 @@ This gives a ring homomorphism
\]
\begin{remark}
This can be generalised \emph{drastically} to generalised cohomology theories! This is the arena of the Atiyah--Hirzebruch spectral sequence.
\begin{itemize}
\item For the purpose of the story told here, we mainly need to remember the Chern character as a construction, that takes us from \K-theory to cohomology.
This is arguably quite ad-hoc.
The Chern character can however be generalised \emph{drastically} to complex oriented cohomology theories!
There is also some connection with formal group laws.
\todo[inline]{look into this and maybe connect with Leon's talk, reference: \url{https://mathoverflow.net/questions/6144/explanation-for-the-chern-character}}
% This is the arena of the Atiyah--Hirzebruch spectral sequence.
\item One can also use the differential geometry approach from Chern--Weil theory to define the Chern character.
\end{itemize}
\end{remark}
\todo[inline]{\textcite[Thm.~III.13.13]{lawson_spin} is the closest thing to the Atiyah--Singer formula in \cite{hirzebruch_modularforms} I found so far.}
......
......@@ -10,7 +10,7 @@
contents/characteristic,
contents/elliptic-genera,
% contents/atiyah_singer,
% contents/coarse,
contents/coarse,
contents/appendix
}
......
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