Commit fd76af4d by 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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