%% This LaTeX-file was created by <jl> Mon Mar  3 15:52:55 1997
%% LyX 0.10 (C) 1995 1996 by Matthias Ettrich and the LyX Team

%% Don't edit this file unless you are sure what you are doing.
\documentclass[12pt,letterpaper,oneside,onecolumn]{article}
\usepackage[]{fontenc}
\usepackage[dvips]{epsfig}

%%
%% BEGIN The lyx specific LaTeX commands.
%%

\makeatletter
\def\LyX{L\kern-.1667em\lower.25em\hbox{Y}\kern-.125emX\spacefactor1000}%
\newcommand{\lyxtitle}[1] {\thispagestyle{empty}
\global\@topnum\z@
\section*{\LARGE \centering \sffamily \bfseries \protect#1 }
}
\newcommand{\lyxline}[1]{
{#1 \vspace{1ex} \hrule width \columnwidth \vspace{1ex}}
}
\newenvironment{lyxbibliography}
{
\begin{thebibliography}{99}}
{\end{thebibliography}}
\def\lxq{"}
\newenvironment{lyxcode}
{\list{}{
\rightmargin\leftmargin
\raggedright
\itemsep 0pt
\parsep 0pt
\ttfamily
}%
\item[]
}
{\endlist}
\newcommand{\lyxlabel}[1]{#1 \hfill}
\newenvironment{lyxlist}[1]
{\begin{list}{}
{\settowidth{\labelwidth}{#1}
\setlength{\leftmargin}{\labelwidth}
\addtolength{\leftmargin}{\labelsep}
\renewcommand{\makelabel}{\lyxlabel}}}
{\end{list}}
\newcommand{\lyxletterstyle}{
\setlength\parskip{0.7em}
\setlength\parindent{0pt}
}
\newcommand{\lyxaddress}[1]{
\par {\raggedright #1 
\vspace{1.4em}
\noindent\par}
}
\newcommand{\lyxrightaddress}[1]{
\par {\raggedleft \begin{tabular}{l}\ignorespaces
#1
\end{tabular}
\vspace{1.4em}
\par}
}
\newcommand{\lyxformula}[1]{
\begin{eqnarray*}
#1
\end{eqnarray*}
}
\newcommand{\lyxnumberedformula}[1]{
\begin{eqnarray}
#1
\end{eqnarray}
}
\makeatother

%%
%% END The lyx specific LaTeX commands.
%%

\pagestyle{plain}
\setcounter{secnumdepth}{3}
\setcounter{tocdepth}{3}
\begin{document}

holomorphic vector bundle= vector bundle over a complex manifold with
a connection that has holomorphic transition functions up to equivalence
under \( GL(k,C) \) transition functions.

\( (v_{i})_{a}=(t_{ij}(z))_{ab}(v_{j})_{b} \)

\( E(M,V_{p}) \): holomorphic principle bundle representation

Fiber is \( V_{p}=F=C^{k} \) 

consider sections of \( E(M,V_{p}) \), \( f:M\rightarrow E \)

f is holomorphic iff \( \nabla _{u^{*}}f=0 \).

necessary condition for existence of f is \( [\nabla _{u^{*}},\nabla _{v^{*}}]f=0=F_{u^{*}v^{*}}f=0 \) This means \( F_{u^{*}v^{*}}=0 \) in other
words, a portion of the curvature must be 0.

curvature = 0 for the antiholomorphic part means \( A_{u^{*}}=V^{-1}\partial _{u^{*}}V \) with \( V\in GL(k,C) \), not necessarily
in G.

If you generalize gauge to \( GL(k,C) \) then you can choose: \( A_{u^{*}}=0 \) and \( A_{u}\neq 0 \).

Then the covariant derivatives: \( \nabla =D+D^{*} \), satisfy: 

\begin{itemize}

\item \( D=\partial +A \)

\item \( D^{*}=\partial ^{*} \)

\end{itemize}

Then \( F=(\partial +\partial ^{*})(A+A^{*})+(A+A^{*})\wedge (A+A^{*}) \) simplifies too \( F=(\partial +\partial ^{*})A+A\wedge A \) .

Example of holomorphic vector bundle: tangent bundles

Theorem: holomorphic principle bundle, P(M,G), then for every representation
p of G, \( \exists  \) an associated holomorphic vector bundle \( E(m,V_{p}) \).

To prove this, relate \( A_{u^{*}} \) to \( A_{v^{*}} \) via transition functions. Then derive
that \( \partial _{u^{*}}A_{u^{*}}=0 \).

To holomorphic transition functions \( \{t_{ij}\} \) and \( \{t_{ij}'\} \) are equivalent iff \( t_{ij}'=V^{(i)}t_{ij}V^{(j)-1} \) where
\( V^{(i)} \) are holomorphic. 



\end{document}
