
\input amstex

\documentstyle{amsppt}

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\define\GL{\operatorname{GL}}
\define\ORTH{\operatorname{O}}
\define\U{\operatorname{U}}
\define\Lie{\operatorname{Lie}}

\define\Hom{\operatorname{Hom}}
\define\Iso{\operatorname{Iso}}
\define\End{\operatorname{End}}
\define\Aut{\operatorname{Aut}}
\define\Ker{\operatorname{Ker}}
\define\Coker{\operatorname{Coker}}
\redefine\Im{\operatorname{Im}}

\define\Rank{\operatorname{Rank}}
\define\Tr{\operatorname{Tr}}
\define\Mat{\operatorname{Mat}}

\define\rp{\Bbb{RP}}
\define\cp{\Bbb{CP}}

\define\R{\Bbb R}
\define\C{\Bbb C}
\define\Z{\Bbb Z}
\redefine\P{\Bbb P}

\hsize  6.5 true in
\hoffset  0.0 true in
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\NoRunningHeads
\parindent 16 pt



\title       The Geometry of Vector Bundles and an Introduction to Gauge
Theory \\
Lecture 30
\endtitle
\author      Professor Steven Bradlow \\ Class Notes From Math 433
\endauthor
\affil       University of Illinois at Urbana-Champaign
\endaffil
\address     273 Altgeld Hall, 1409 W. Green Street, Urbana, IL  61801
\endaddress
\email       bradlow\@math.uiuc.edu
\endemail
\date        April 13, 1998
\enddate

\endtopmatter


\document

\vskip.25in
\leftline{{\bf Characteristic Classes for Complex Bundles II}}
\vskip.25in




\definition{Definition 1} Let $E\to B$ be a complex bundle of rank $n$.
To define the {\it Chern classes} of $E$ we use the
($\GL(n,\C)$-invariant) polynomials corresponding to the homogeneous
parts of $\det(I+A)$, i.e. we use $c_j(A)$ where 
$$\det(I+A)=\sum^n_{j=0}c_j(A).$$ 
We define the {\it j-th Chern class} of $E$ by
$$c_j(E)=c_j(\frac{i}{2\pi}F)\ \in H^{2j}(B,\C),$$
where $F$ is the curvature form of any connection on $E$, expressed
with respect to any local frame.
We define the {\it total Chern class} of $E$ by
$$c(E)=\sum^n_{j=0}c_j(E)=\det(I+\frac{i}{2\pi}F).$$
\enddefinition

\remark{Note} We can always
\itemitem{$\bullet$} fix \underbar{Hermitian metric} on $E$ 
\itemitem{$\bullet$} take $D$ \underbar{unitary}
\itemitem{$\bullet$} use \underbar{unitary frames}

Then $F+F^*=0$. (Thus if $F+F_{ij}\,dx_i\wedge dx_j$ with respect to
local coordinates, then $F_{ij}+F_{ij}^*=0$.)
\endremark

\vskip.12in
\noindent
\underbar{Recall}: Any $A\in\U(n)=\Lie(\U(n))$ is diagonalizable by
$T\in\U(n)$, so we can restrict our attentation to 
$$\boxed{\text{$\U(n)$-invariant polynomials on $\U(n)$}}$$

\proclaim{Claim} The Chern classes are real cohomology classes, i.e.
$c_j(E)\in H^{2j}(B,\R)$.
\endproclaim

\noindent
That is, given any $X_1,\cdots,X_{2j}\in\Omega^0(TB)$, 
$c_j(E)(X_1,\cdots,X_{2j})\in\R$.
(In fact, the classes are {\bf integral}, i.e. $c_j(E)\,[\sigma]\in\Z$ for
$[\sigma]\in H_{2j}(B,\Z)$, but we won't prove that here.)

\vskip.08in
{\bf Proof:} If $F+F^*=0$, then $\frac{i}{2\pi}F=(\frac{i}{2\pi}F)^*$. 
Thus $\det(I+\frac{i}{2\pi}F)=\overline{\det(I+\frac{i}{2\pi}F)}$.
Therefore $c_j(E)=\overline{c_j(E)}$, for all $j$. (Explicitiy: if 
$c_j(E)=\left[ \,\sum\gamma_{_I}\,dx_{i_1}\wedge\cdots\wedge
dx_{i_{2j}}\,\right]$,
then $\gamma_{_I}=\overline{\gamma_{_I}}$.)\qed 

\example{Example 1} For a line bundle $\gamma\to\P^1$ (recall
$\gamma|_{[z_0,z_1]}=\text{line through $(z_0,z_1)$}$),
$c_0(\gamma)=\det(I)=1$ and
$c_1(\gamma)=\Tr(\frac{i}{2\pi}F)=\frac{i}{2\pi}F\in H^2(\P^1,\R)$. 
(In general, 
$c_j(E)$ is defined for $j\leq\text{min}(\frac{\text{dim}B}{2},\Rank(E))$.)
Then $$\int_{\P^1}\,c_1(\gamma)=c_1(\gamma)\,[\P^1]=-1$$ (we will see
the proof later\,!) 
\endexample

Before we get to more computations, we look at general properties:

\noindent
{\bf Formulae}\,: From definition and expression for
$\det(I+\frac{i}{2\pi}A)=\sum c_j(A)$, we can see
$$\align
  c_0(A) &=\det I=1\\
  c_1(A) &=\Tr(\frac{i}{2\pi}A)\\
  \vdots \quad&\\
  c_n(A) &=\det(A)
  \endalign$$
[ \quad if we do the computation for $I+\frac{i}{2\pi}A$ 
\underbar{diagonal}, and we write the diagonal matrix for 
$I+\frac{i}{2\pi}A$ as
$$
\left( \matrix
      1+x_1& & \\
           &\ddots& \\
       & &1+x_n
\endmatrix \right)
$$
\quad\ \   
where $x_i=\frac{i}{2\pi}\lambda_i$ and $\lambda_i$ are eigenvalues of
$A$, then 
$$\qquad\qquad\qquad\quad\det(I+\frac{i}{2\pi}A)
=\prod^n_{i=1}(1+x_i)
=1+\sum x_i+\sum_{i\neq j}x_ix_j+\cdots+\prod^n_{i=1}x_i
\qquad\qquad\qquad\qquad]
$$   

\vskip.12in
Sometimes more convenient to define the {\bf Chern characters}:
$$ch_k(E)=\frac{1}{k!}\,\Tr(\frac{i}{2\pi}F)^k\qquad k=0,1,2,\cdots.$$
Also, the {\bf Total Chern character} is defined by
$$\align
  ch(E)&=\sum_{k=0}^{\infty}ch_k(E)\\
       &=\sum_{k=0}^{\infty}\frac{1}{k!}\,\Tr(\frac{i}{2\pi}F)^k\\
       &=\Tr(\ \sum_{k=0}^{\infty}\frac{(\frac{i}{2\pi}F)^k}{k!}\ )\\
       &=\Tr(\,\exp(\frac{i}{2\pi}F)\,).
  \endalign$$

\proclaim{Property 1} $c(E_1\oplus E_2)=c(E_1)\cdot c(E_2)\qquad
  (\ \text{in \ $H^*(B,\R)$}\ )$.  
\endproclaim
{\bf Proof}: Let's pick connections $D_i$ on $E_i$, $i=1,2$.
Then we construct $D=D_1\oplus D_2$. With respect to local frames, we get
$$A=\left( \matrix
           A_1&0\\
           0&A_2
           \endmatrix\right)
   \quad \and\quad
  F=\left( \matrix
           F_1&0\\
           0&F_2
           \endmatrix\right).$$
Therefore
$$\align c(E)&=c(E_1\oplus E_2)\\
             &=\det\left(I+\frac{i}{2\pi}F\right)\\
             &=\det\left( \bmatrix
                          I_1+\frac{i}{2\pi}F_1&0\\
                           0&I_2+\frac{i}{2\pi}F_2\endbmatrix
                   \right)\\
             &=\det\left(I_1+\frac{i}{2\pi}F_1\right)\cdot
                \det\left(I_2+\frac{i}{2\pi}F_2\right)\\
             &=c(E_1)\cdot c(E_2)
   \endalign$$
\qed

\proclaim{Corollary} We can extract formulae for $c_j(E)$ from
$$\sum^{n_1+n_2}_{j=0}c_j(E)=
\left(\sum^{n_1}_{i=0}c_i(E_1)\right)\cdot\left(\sum^{n_2}_{k=0}c_k(E_2)\right).$$
\endproclaim
For example, 
$$
\alignat2
\text{deg 1}&:\qquad c_1(E)&&=c_1(E_2)+c_1(E_1)\\
\text{deg 2}&:\qquad c_2(E)&&=c_2(E_1)+c_1(E_1)c_1(E_2)+c_2(E_2)\\ 
\vdots&\\
\text{etc.} &         
\endalignat
$$

\proclaim{Property 2} $ch(E_1\oplus E_2)=ch(E_1)+ch(E_2)$, hence 
$ch_k(E_1\oplus E_2)=ch_k(E_1)+ch_k(E_2)$.
\endproclaim 
{\bf Proof}: Use $D=D_1+D_2$ and 
$$
\align
\Tr(\exp(\frac{i}{2\pi}F))
&=\Tr\left(\bmatrix \exp(\frac{i}{2\pi}F_1)&0\\
                   0&\exp(\frac{i}{2\pi}F_2)\endbmatrix\right)\\
&=\Tr(\exp(\frac{i}{2\pi}F_1))+
  \Tr(\exp(\frac{i}{2\pi}F_2)).
\endalign $$
\qed 



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