Last update: 28 January 2012

In this section we specify the relations between the different aspects and the different usages of the notion of "local system of complex vector spaces". The equivalence between the different points of view has been well known for a long time.

The "crystalline" point of view has not been considered, see [4], [10].

Let $X$ be a topological space. A **complex local system** on $X$ is a sheaf of complex vector spaces on $X$ which, locally on $X$, are isomorphic to one of the constant sheaves
${\u2102}^{n}\left(n\in {\mathbb{Z}}_{>0}\right).$

Let $X$ be a locally path connected and locally path simply connected topological space with a base point ${x}_{0}\in X.$ To avoid all ambiguity we specify that

- The fundamental group
${\pi}_{1}\left(X,{x}_{0}\right)$
of $X$ at
${x}_{0},$
has as elements the homotopy equivalence classes of loops
**word cut off** - If $\alpha ,\beta \in {\pi}_{1}\left(X,{x}_{0}\right)$ are represented by loops $a$ and $b$, then $\alpha \beta $ is represented by the loop $ab$ obtained by juxtaposing $b$ and $a$, in that order.

Under the hypotheses 1.2, with $X$ connected, the functor $\mathcal{F}\to {\mathcal{F}}_{{x}_{0}}$ is an equivalence between the category of locally constant sheaves on $X$ and the category of sets with an action of the group ${\pi}_{1}\left(X,{x}_{0}\right).$

Under the hypotheses of 1.2, with $X$ connected, the functor $\mathcal{F}\to {\mathcal{F}}_{{x}_{0}}$ is an equivalence between the category of local systems on $X$ and the category of complex finite dimensional representations of ${\pi}_{1}\left(X,{x}_{0}\right).$

Under the hypotheses 1.2, if $a: [0,1] \to X$ is a path, and $b$ is a loop at $a\left(0\right)$ then $ab{a}^{-1}=a\left(b\right)$ is a loop at $a\left(1\right).$ Its homotopy class only depends on $a$ and $b$. This construction defines an isomorphism between ${\pi}_{1}\left(X,a\left(0\right)\right)$ and ${\pi}_{1}\left(X,a\left(1\right)\right).$

Under the hypotheses 1.5, there exists up to unique isomorphism a unique locally constant sheaf of groups on $X$ (the fundamental groupoid), with for each $x\in X,$ an isomorphism $$\begin{array}{cc}\text{(1.6.1)}& {\pi}_{1}{\left(X\right)}_{{x}_{0}}\simeq {\pi}_{1}\left(X,{x}_{0}\right)\end{array}$$ such that, for each path $a: [0,1] \to X,$ the isomorphism in 1.5 between ${\pi}_{1}\left(X,a\left(0\right)\right)$ and ${\pi}_{1}\left(X,a\left(1\right)\right)$ is identified via (1.6.1) with the isomorphism between ${\pi}_{1}{\left(X\right)}_{a\left(0\right)}$ and ${\pi}_{1}{\left(X\right)}_{a\left(1\right)}.$ If $X$ is connected with base point ${x}_{0},$ the sheaf ${\pi}_{1}\left(X\right)$ corresponds, via the equivalence 1.3, to the group ${\pi}_{1}\left(X,{x}_{0}\right)$ with its action on itself by inner automorphisms.

If $\mathcal{F}$ is a locally constant sheaf on $X$ then there exists a unique action (called **canonical**) of
${\pi}_{1}\left(X\right)$
on $\mathcal{F}$ such that in each
${x}_{0}\in X$
induces the action 1.2 of
${\pi}_{1}\left(X,{x}_{0}\right)$
on $\mathcal{F}.$

Let $X$ be an analytic space (0.1). We will call a **vector bundle** (holomorphic) on $X$ a sheaf of locally free modules of finite type over the structure sheaf $\mathcal{O}$ of $X$. If $\mathcal{V}$ is a vector bundle on $X$ and $x$ is a point of $X$, we will write
${\mathcal{V}}_{\left(x\right)}$
for the free
${\mathcal{O}}_{\left(x\right)}-$module of finite type of germs of sections of $\mathcal{V}$. If
${\U0001d52a}_{x}$
is the maximal ideal of
${\mathcal{O}}_{\left(x\right)},$
we will call the **fiber in $x$ of the vector bundle $\mathcal{V}$** the $\u2102-$vector space of finite rank
$$\begin{array}{cc}\text{(2.1.1)}& {\mathcal{V}}_{x}={\mathcal{V}}_{\left(x\right)}{\otimes}_{{\mathcal{O}}_{\left(x\right)}}\frac{{\mathcal{O}}_{\left(x\right)}}{{\U0001d52a}_{x}}\end{array}$$
If
$f:X\to Y$
is a morphism of analytic spaces, the inverse image vector bundle
${f}^{*}\mathcal{V}$
on $X$ of a vector bundle $\mathcal{V}$ on $Y$ is the inverse image of $\mathcal{V}$ as a coherent module: if
${f}^{\circ}\mathcal{V}$
is the sheaf theoretic inverse image of $\mathcal{V}$, one has
$$\begin{array}{cc}\text{(2.1.2)}& {f}^{*}=\end{array}$$
**eqn cut off**

In particular, if
$x:P\to X$
is the morphism of the one point space into $X$ defined by a point $x$ of $X$, one has
$$\begin{array}{cc}\text{(2.1.3)}& {\mathcal{V}}_{x}\simeq {x}^{*}\mathcal{V}.\end{array}$$

Let $X$ be a complex analytic variety (0.7) and $\mathcal{V}$ a complex vector bundle on $X.$ Historically one would define a (holomorphic) connection on $\mathcal{V}$ by the following data: for each pair of points infinitely close to 1^{st} order
$\left(x,y\right)$
of $X$, an isomorphism
${\gamma}_{y,x}:{\mathcal{V}}_{x}\to {\mathcal{V}}_{y},$
this isomorphism depends holomorphically on
$\left(x,y\right)$
and satisfies
${\gamma}_{x,x}=\mathrm{id}.$

If this is interpreted appropriately, this "definition" coincides with the definition which we now give in 2.2.4 below (which will not be used in the rest of this section).

It suffices for our purposes to interpret "point" as signifying "a point with value in some analytic space".

**A point of an analytic space** $X$ **has value in an analytic space** $S$ is a morphism from $S$ to $X.$

If $Y$ is a subspace of $X$, **the n ^{th} infinite neighborhood** of $Y$ in $X$ is the subspace of $X$ defined locally by the (n+1)

Two points $x$, $y$ of $X$ having values in $S$ are called 1^{st} order infinitesimal neighbors if the function
$\left(x,y\right):S\to X\times X$
which they define factors through a 1^{st} order infinitesimal neighborhood of the diagonal of
$X\times X.$

If $X$ is a complex analytic variety and $\mathrm{Vscr;}$ is a vector bundle on $X,$ a (**holomorphic**) **connection** $\gamma $ on $\mathcal{V}$ consists of the following data:

- for every pair
$\left(x,y\right)$
of points of $X$ having values in some analytic space $S$, with $x$ and $y$ 1
^{st}order infinitesimal neighbors, we are given ${\gamma}_{x,y}:{x}^{*}\mathcal{V}\to {y}^{*}\mathcal{V}:$ this data is subject to the conditions:- (functoriality) If there is
$f:T\to S$
and points which are 1
^{st}order infinitesimal neighbors $x,y:S\overrightarrow{\to}X$ then ${f}^{*}\left({\gamma}_{y,x}\right)={\gamma}_{yf,xf}.$ - One has ${\gamma}_{x,x}=\mathrm{id}.$

- (functoriality) If there is
$f:T\to S$
and points which are 1

Let
${X}_{1}$
be the 1^{st} order infinitesimal neighborhood of the diagonal
${X}_{0}$
of
$X\times X$
and let
${p}_{1}$
and
${p}_{2}$
be the two projections of
${X}_{1}$
on $X$. By definition, the vector bundle
${\mathcal{P}}^{1}\left(\mathcal{V}\right)$
of jets of 1^{st} order sections of $\mathcal{V}$ is the fiber
${{p}_{1}}_{*}{{p}_{2}}^{*}\mathcal{V}.$
We will denote by
${j}^{1}$
the 1^{st} order differential operator which to each section of $\mathcal{V}$ associates its 1^{st} order jet:
$${j}^{1}:\mathcal{V}\to {\mathcal{P}}^{1}\left(\mathcal{V}\right)\simeq {\mathcal{O}}_{{X}_{1}}{\otimes}_{{\mathcal{O}}_{X}}\mathcal{V}.$$
A connection 2.2.4 can be interpreted as a homomorphism (automatically an isomorhpism)
$$\begin{array}{cc}\text{(2.3.1)}& \gamma :{{p}_{1}}^{*}\mathcal{V}\to {{p}_{2}}^{*}\mathcal{V}\end{array}$$
which induces the identity over
${X}_{0}.$
Since
$${\mathrm{Hom}}_{{X}_{1}}\left({{p}_{1}}^{*}\mathcal{V},{{p}_{2}}^{*}\mathcal{V}\right)=\mathrm{Hom}\left(\mathcal{V},{{p}_{1}}_{*}{{p}_{2}}^{*}\mathcal{V}\right),$$
a connection can be interpreted again as a homomorphism ($\mathcal{O}-$linear)
$$\begin{array}{cc}\text{(2.3.2)}& D:\mathcal{V}\to {\mathcal{P}}^{1}\left(\mathcal{V}\right)\end{array}$$
such that the composition
$$\mathcal{V}\stackrel{D}{\to}{\mathcal{P}}^{1}\left(\mathcal{V}\right)\to \mathcal{V}$$
is the identity. The sections
$D\left(s\right)$
and
${j}^{1}\left(s\right)$
of
${\mathcal{P}}^{1}\left(\mathcal{V}\right)$
thus have the same image in $\mathcal{V}$ and
${j}^{1}\left(s\right)-D\left(s\right)$
identifies with a section
$\nabla s$
of
${\Omega}_{X}^{1}\otimes \mathcal{V}\simeq \mathrm{ker}\left({\mathcal{P}}^{1}\left(\mathcal{V}\right)\to \mathcal{V}\right):$
$$\begin{array}{cc}\text{(2.3.3)}& \nabla :\mathcal{V}\to {\Omega}^{1}\left(\mathcal{V}\right)\\ \text{(2.3.4)}& {j}^{1}\left(s\right)=D\left(s\right)+\nabla \left(s\right).\end{array}$$
In other words a connection permits us to compare two neighboring fibers of $\mathcal{V}$, and also permits the definition of the differential
$\nabla s$
of a section of $\mathcal{V}.$

Conversely, the formula 2.3.4 permits one to define $D$ and thus $\gamma $ in terms of the covariant derivative $\nabla $. For $D$ to be linear it is necessary and sufficient that $\nabla $ satisfies the identity $$\begin{array}{cc}\text{(2.3.5)}& \nabla \left(fs\right)=df\cdot s+f\cdot \nabla s.\end{array}$$ The definition 2.2.4 is equivalent to the following definition, due to J.L. Koszul.

Let $\mathcal{V}$ be a holomorphic vector bundle on a complex analytic variety $X$. A holomorphic connection (of simply a connection) on $\mathcal{V}$ is a $\u2102-$linear homomorphism
$$\nabla :\mathcal{V}\to {\Omega}_{X}^{1}\left(\mathcal{V}\right)={\Omega}_{X}^{1}{\otimes}_{\mathcal{O}}\mathcal{V}$$
satisfying the **Leibniz identity** (2.3.5) for $f$ and $s$ local sections of $\mathcal{O}$ and $\mathcal{V}$. We call $\nabla $ the covariant derivative defined by the connection.

If the vector bundle $\mathcal{V}$ is endowed with a connection $\Gamma $ with covariant derivative $\nabla $ and if $w$ is a holomorphic vector field on $X$, we put, for each local section $v$ of $\mathcal{V}$ on an open set $U$ of $X$
$${\nabla}_{w}\left(v\right)= \u27e8\nabla v,w\u27e9 \in \mathcal{V}\left(U\right).$$
We call
${\nabla}_{w}:\mathcal{V}\to \mathcal{V}$
the **covariant derivative along the vector field $w$**.

If ${}_{1}\Gamma $ and ${}_{2}\Gamma $ are two connections on $X$, with covariant derivatives ${}_{1}\nabla $ and ${}_{2}\nabla ,$ then ${}_{2}\nabla -{}_{1}\nabla $ is an $\mathcal{O}-$linear homomorphism from $\mathcal{V}$ to ${\Omega}_{X}^{1}\left(\mathcal{V}\right).$ Conversely, the sum of ${}_{1}\nabla $ and such a homomorphism defines a connection on $\mathcal{V}$: the connections on $\mathcal{V}$ form a principle homogeneous space (or torsor) on $\underset{\_}{\mathrm{Hom}}\left(\mathcal{V},{\Omega}_{X}^{1}\left(\mathcal{V}\right)\right)\simeq {\Omega}_{X}^{1}\left(\underset{\_}{\mathrm{End}}\left(\mathcal{V}\right)\right).$

Let $X$ be a complex analytic variety. If $\mathcal{V}$ is the holomorphic vector bundle defined by a $\u2102-$vector space
${V}_{0},$
we have seen that $\mathcal{V}$ admits a canonical connection of covariant derivative
${}_{0}\nabla .$
If $\nabla $ is the covariant derivative defined by another connection on $\mathcal{V}$, we have seen (2.6) that $\nabla $ can be written in the form
$$\nabla ={}_{0}\nabla +\Gamma ,\phantom{\rule{2em}{0ex}}with\phantom{\rule{.5em}{0ex}}\Gamma \in \Omega \left(\underset{\_}{\mathrm{End}}\left(\mathcal{V}\right)\right).$$
If we identify sections of $\mathcal{V}$ and holomorphic maps from $X$ into
${V}_{0}$
then one has
$$\begin{array}{cc}\text{(3.1.1)}& \nabla v=dv+\Gamma \cdot v\end{array}.$$
If one fixes a basis of $\mathcal{V}$, i.e. an isomorphism
${\u2102}^{n}\to {V}_{0}$
with coordinates (identified with the basis vectors)
${e}_{\alpha}:\u2102\to {V}_{0},$
then $\Gamma $ is represented as a matrix of differential forms
${\omega}_{\beta}^{\alpha}$
(the matrix of forms of the connection), and (3.1.1) can be rewritten
$$\begin{array}{cc}\text{(3.1.2)}& {\left(\nabla v\right)}^{\alpha}=d{v}^{\alpha}+\sum _{\beta}{\omega}_{\beta}^{\alpha}{v}^{\beta}.\end{array}$$
Let $\mathcal{V}$ be a holomorphic vector bundle on $X$. The choice of a base
$e:{\u2102}^{n}\stackrel{\sim}{\to}\mathcal{V}$
of $\mathcal{V}$ permits us to consider $\mathcal{V}$ as defined by a constant vector bundle
$\left({\u2102}^{n}\right),$
and the preceding conditions apply: the connections on $\mathcal{V}$ correspond, via (3.1.2.), with
$n\times n$
matrices of differential forms on $X$. If
${\omega}_{e}$
is the matrix of the connection $\nabla $ in the basis $e$, and if
$f:{\u2102}^{n}\stackrel{\sim}{\to}\mathcal{V}$
is a new basis of $\mathcal{V}$, with coordinates
$A\in {\mathrm{GL}}_{n}\left(\mathcal{O}\right),\phantom{\rule{.5em}{0ex}}\left(A={e}^{-1}f\right),$
one has (3.1.2)
$$\begin{array}{rcl}\nabla v& =& ed\left({e}^{-1}v\right)+e{\omega}_{e}{e}^{-1}v\\ & =& f{A}^{-1}d\left(A{f}^{-1}v\right)+f{A}^{-1}{\omega}_{e}A{f}^{-1}v\\ & =& fd{f}^{-1}v+f\left({A}^{-1}dA+{A}^{-1}{\omega}_{e}A\right){f}^{-1}v.\end{array}$$
Comparing with (3.1.2) in the basis $f$, one finds that
$$\begin{array}{cc}\text{(3.1.3)}& {\omega}_{f}={A}^{-1}dA+{A}^{-1}{\omega}_{e}A.\end{array}$$
If also
$\left({x}^{i}\right)$
is a system of local coordinates on $X$, defining a basis of
${\Omega}_{X}^{1}$
of basis vectors
$d{x}^{i},$
we put
$${\omega}_{\beta}^{\alpha}=\sum _{i}\Gamma _{\beta}{}_{i}{}^{\alpha}d{x}^{i}$$
and we call the holomorphic functions
$\Gamma _{\beta}{}_{i}{}^{\alpha}$
the **coefficients of the connection.** The formula 3.1.2 becomes
$${\left({\nabla}_{i}v\right)}^{\alpha}={\partial}_{i}{v}^{\alpha}+\sum _{\beta}\Gamma _{\beta}{}_{i}{}^{\alpha}{v}^{\beta}.$$
The differential equation
$\nabla v=0$
for horizontal sections of $\mathcal{V}$ is written as a system of 1^{st} order partial differential equations, linear and homogeneous
$$\begin{array}{cc}\text{(3.1.5)}& {\partial}_{i}{v}^{\alpha}=-\sum _{\beta}\Gamma _{\beta}{}_{i}{}^{\alpha}{v}^{\beta}.\end{array}$$

With the notations of (3.1.2), and using the convention of blind summation indices, one has $$\begin{array}{rcl}\nabla \nabla v& =& \nabla \left(\left(d{v}^{\alpha}+{\omega}_{\beta}^{\alpha}{v}^{\beta}\right){e}_{\alpha}\right)\\ & =& d\left(d{v}^{\alpha}+{\omega}_{\beta}^{\alpha}{v}^{\beta}\right){e}_{\alpha}-\left(d{v}^{\alpha}+{\omega}_{\beta}^{\alpha}{v}^{\beta}\right)\wedge {\omega}_{\alpha}^{\gamma}{e}_{\gamma}\\ & =& d{\omega}_{\beta}^{\alpha}\cdot {v}^{\beta}\cdot {e}_{\alpha}-{\omega}_{\beta}^{\alpha}\wedge d{v}^{\beta}\cdot {e}_{\alpha}-d{v}^{\alpha}\wedge {\omega}_{\alpha}^{\gamma}\cdot {e}_{\gamma}-{\omega}_{\beta}^{\alpha}\wedge {\omega}_{\alpha}^{\gamma}\cdot {v}^{\beta}{e}_{\gamma}\\ & =& \left(d{\omega}_{\beta}^{\gamma}-{\omega}_{\beta}^{\alpha}\wedge {\omega}_{\alpha}^{\gamma}\right){v}^{\beta}{e}_{\gamma}.\end{array}$$ The matrix of the curvature tensor is thus $$\begin{array}{cc}\text{(3.2.1)}& {R}_{\beta}^{\alpha}=d{\omega}_{\beta}^{\alpha}+\sum _{\gamma}{\omega}_{\gamma}^{\alpha}\wedge {\omega}_{\beta}^{\gamma},\end{array}$$ a formula which we write also as $$\begin{array}{cc}\text{(3.2.2)}& R=d\omega +\omega \wedge \omega .\end{array}$$ The formula 3.2.1 provides, in a system of local coordinates $\left({x}^{i}\right)$ $$\begin{array}{rcl}R_{\beta ,}^{\alpha}{}_{i,}{}_{j}& =& \left({\partial}_{i}\Gamma _{\beta}{}_{j}{}^{\alpha}-{\partial}_{j}\Gamma _{\beta}{}_{i}{}^{\alpha}\right)+\left(\Gamma _{\gamma}{}_{i}{}^{\alpha}\Gamma _{\beta}{}_{j}{}^{\gamma}-\Gamma _{\gamma}{}_{j}{}^{\gamma}\Gamma _{\beta}{}_{i}{}^{\gamma}\right)\\ {R}_{\beta}^{\alpha}& =& \sum _{i<j}R_{\beta ,}^{\alpha}{}_{i,}{}_{j}d{x}^{i}\wedge d{x}^{j}\end{array}$$ The condition $R_{\beta ,}^{\alpha}{}_{i,}{}_{j}=0$ is the condition for integrability of the system (3.1.5) in the classical sense of the word; it can be obtained by eliminating ${v}^{\alpha}$ from the equations obtained by substituting (3.1.5) in the identity ${\partial}_{i}{\partial}_{j}{v}^{\alpha}={\partial}_{j}{\partial}_{i}{v}^{\alpha}.$