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 , .Need to include the references from the original paper here.
Let be a topological space. A complex local system on is a sheaf of complex vector spaces on which, locally on , are isomorphic to one of the constant sheaves
Let be a locally path connected and locally path simply connected topological space with a base point To avoid all ambiguity we specify that
Under the hypotheses 1.2, with connected, the functor is an equivalence between the category of locally constant sheaves on and the category of sets with an action of the group
Under the hypotheses of 1.2, with connected, the functor is an equivalence between the category of local systems on and the category of complex finite dimensional representations of
Under the hypotheses 1.2, if is a path, and is a loop at then is a loop at Its homotopy class only depends on and . This construction defines an isomorphism between and
Under the hypotheses 1.5, there exists up to unique isomorphism a unique locally constant sheaf of groups on (the fundamental groupoid), with for each an isomorphism such that, for each path the isomorphism in 1.5 between and is identified via (1.6.1) with the isomorphism between and If is connected with base point the sheaf corresponds, via the equivalence 1.3, to the group with its action on itself by inner automorphisms.
If is a locally constant sheaf on then there exists a unique action (called canonical) of on such that in each induces the action 1.2 of on
Let be an analytic space (0.1). We will call a vector bundle (holomorphic) on a sheaf of locally free modules of finite type over the structure sheaf of . If is a vector bundle on and is a point of , we will write for the free module of finite type of germs of sections of . If is the maximal ideal of we will call the fiber in of the vector bundle the vector space of finite rank If is a morphism of analytic spaces, the inverse image vector bundle on of a vector bundle on is the inverse image of as a coherent module: if is the sheaf theoretic inverse image of , one has eqn cut off In particular, if is the morphism of the one point space into defined by a point of , one has
Let be a complex analytic variety (0.7) and a complex vector bundle on Historically one would define a (holomorphic) connection on by the following data: for each pair of points infinitely close to 1st order of , an isomorphism this isomorphism depends holomorphically on and satisfies
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 has value in an analytic space is a morphism from to
If is a subspace of , the nth infinite neighborhood of in is the subspace of defined locally by the (n+1)st power of the ideal in which defines
Two points , of having values in are called 1st order infinitesimal neighbors if the function which they define factors through a 1st order infinitesimal neighborhood of the diagonal of
If is a complex analytic variety and is a vector bundle on a (holomorphic) connection on consists of the following data:
Let be the 1st order infinitesimal neighborhood of the diagonal of and let and be the two projections of on . By definition, the vector bundle of jets of 1st order sections of is the fiber We will denote by the 1st order differential operator which to each section of associates its 1st order jet: A connection 2.2.4 can be interpreted as a homomorphism (automatically an isomorhpism) which induces the identity over Since a connection can be interpreted again as a homomorphism (linear) such that the composition is the identity. The sections and of thus have the same image in and identifies with a section of In other words a connection permits us to compare two neighboring fibers of , and also permits the definition of the differential of a section of
Conversely, the formula 2.3.4 permits one to define and thus in terms of the covariant derivative . For to be linear it is necessary and sufficient that satisfies the identity The definition 2.2.4 is equivalent to the following definition, due to J.L. Koszul.
Let be a holomorphic vector bundle on a complex analytic variety . A holomorphic connection (of simply a connection) on is a linear homomorphism satisfying the Leibniz identity (2.3.5) for and local sections of and . We call the covariant derivative defined by the connection.
If the vector bundle is endowed with a connection with covariant derivative and if is a holomorphic vector field on , we put, for each local section of on an open set of We call the covariant derivative along the vector field .
If and are two connections on , with covariant derivatives and then is an linear homomorphism from to Conversely, the sum of and such a homomorphism defines a connection on : the connections on form a principle homogeneous space (or torsor) on
Let be a complex analytic variety. If is the holomorphic vector bundle defined by a vector space we have seen that admits a canonical connection of covariant derivative If is the covariant derivative defined by another connection on , we have seen (2.6) that can be written in the form If we identify sections of and holomorphic maps from into then one has If one fixes a basis of , i.e. an isomorphism with coordinates (identified with the basis vectors) then is represented as a matrix of differential forms (the matrix of forms of the connection), and (3.1.1) can be rewritten Let be a holomorphic vector bundle on . The choice of a base of permits us to consider as defined by a constant vector bundle and the preceding conditions apply: the connections on correspond, via (3.1.2.), with matrices of differential forms on . If is the matrix of the connection in the basis , and if is a new basis of , with coordinates one has (3.1.2) Comparing with (3.1.2) in the basis , one finds that If also is a system of local coordinates on , defining a basis of of basis vectors we put and we call the holomorphic functions the coefficients of the connection. The formula 3.1.2 becomes The differential equation for horizontal sections of is written as a system of 1st order partial differential equations, linear and homogeneous
With the notations of (3.1.2), and using the convention of blind summation indices, one has The matrix of the curvature tensor is thus a formula which we write also as The formula 3.2.1 provides, in a system of local coordinates The condition is the condition for integrability of the system (3.1.5) in the classical sense of the word; it can be obtained by eliminating from the equations obtained by substituting (3.1.5) in the identity