In this post I will discuss a construction, due to Deligne, for extending a flat vector bundle to a flat logarithmic vector bundle. The setting of the construction is as follows: let be a smooth complex manifold, and let be a divisor (i.e. a codimension 1 subvariety). Then given a branch of the logarithm (it is in fact sufficient to choose a ‘set theoretical’ logarithm) Deligne’s construction gives a canonical way of extending a flat holomorphic vector bundle on the complement of to a holomorphic vector bundle on with a flat logarithmic connection having poles along :
Theorem. Let be a holomorphic vector bundle with flat connection on . There exists a unique extension to all of such that has logarithmic singularities along and the eigenvalues of the residue of lie in the image of .
1. Preliminaries on connections.
Throughout let be a complex manifold, and a holomorphic vector bundle.
- A connection on is a -linear map
satisfying the Liebniz rule: for every function and section
- Given a vector field , we can hook it into the 1-form part of the connection to get a -linear map . The connection is flat if for all vector fields :
Remark. By hooking vector fields into the connection, we see that a connection defines an action of the tangent sheaf of on the vector bundle : . The condition that be flat says simply that this is a morphism of sheaves of Lie algebras. In other words, flat connections are representations of the Lie algebroid .
Locally, we can pick a frame for , allowing us to write a connection as
for a matrix valued 1-form . More precisely let be a local frame. Then , for 1-forms . Then writing a general section in terms of the frame as a column vector we have
It turns out that, despite being defined in terms of a complex structure on and a holomorphic structure on , flat connections are purely topological objects. This is the content of the following theorem:
Theorem. (Riemann-Hilbert Correspondence) There is an equivalence of categories between the category of holomorphic flat connections and the category of finite dimensional representations of the fundamental group of :
The equivalence is given by sending a flat connection to its monodromy representation.
Sketch of Proof. Given a bundle with flat connection , we use the flat sections (i.e. such that ) to identify nearby fibers of . This allows us to define a parallel transport operator , which associates a linear isomorphism to each homotopy class of paths:
Picking a base-point and applying the parallel transport operator to the loops based at then gives the monodromy representation:
Remark. Using the technology of Lie groupoids, this sketch can be turned into a proof: as argued above, a flat connection is a representation of the tangent algebroid. So, using an analogue of Lie’s second theorem for Lie groupoids/algebroids, it can be integrated to a representation of its source simply connected Lie groupoid, which in this case is the fundamental groupoid of
As it turns out, this representation is nothing but the parallel transport operator defined above. Finally, since is transitive, it is Morita equivalent to the fundamental group of (its isotropy group), and therefore we can restrict the parallel transport operator to this group without losing any information. This is summarized in the following two equivalences
The technology of Lie groupoids guarantees that both arrows are equivalences. This perspective is taken up in the following paper.
2. Connections with logarithmic poles.
Let be a divisor which is either smooth or has normal crossing singularities (i.e. it is locally a union of coordinate hyperplanes: there exist coordinates such that ). Let denote the irreducible components of .
- The sheaf of logarithmic tangent vectors is the subsheaf of consisting of vector fields on which are tangent to along . This sheaf is locally free and hence defines a vector bundle . Note also that it inherits a bracket on its sections from the Lie bracket of vector fields. Hence it defines a Lie algebroid, with anchor map given by the inclusion of sheaves.
- Dual to this is the sheaf of 1-forms with logarithmic poles along Y
Given local coordinates such that we have
Proposition. Suppose is smooth. The sheaf of 1-forms with logarithmic poles lies in the following short exact sequence
where is the residue map, given locally by
Remark. In the case that is a divisor with normal crossings we can define a residue map for each component .
- A connection on with logarithmic poles along is a -linear map satisfying the Liebniz rule. The notion of flatness still makes sense in this context. Note that a flat connection with logarithmic poles along defines a representation of the logarithmic tangent sheaf.
- The residue of a logarithmic connection along is the endomorphism defined by composing the connection with the residue on logarithmic forms:
Remark. When is a normal crossing divisor we define the residue for each component .
Given coordinates for and a choice of frame for as above we can write a logarithmic connection as
for holomorphic matrix-valued functions . The residue for this connection is given by .
3. Deligne’s construction for extending connections
Consider the exponential sequence
and choose a set-theoretical splitting . Then we have the following theorem which is due to Deligne:
Theorem. Let be a holomorphic vector bundle, and a holomorphic flat connection on . Then there exists a holomorphic vector bundle with flat logarithmic connection extending to , unique up to unique isomorphism, such that
- has logarithmic poles along ,
- the eigenvalues of the residues of lie in the image of the splitting .
Proof. This proof essentially follows the chapter by Malgrange in the book Algebraic D-modules.
We begin with a simplification: by the uniqueness part of the theorem, it only needs to be proven locally: once extensions are constructed on local neighbourhoods there is a unique way of gluing them together to get a global extension. So assume that is a polydisc:
and that , with ith component . The proof now proceeds in two parts: existence and uniqueness.
is a holomorphic flat connection on and so by the Riemann-Hilbert correspondence it is determined by it’s monodromy representation. First note that since
it follows that . Let us make some choices in order to exhibit this isomorphism: choose a point . Then the fundamental group is generated by loops, and we take the ith loop to be the loop which starts at and goes around counterclockwise in the plane . The monodromy representation is given by taking the parallel transport around each of these loops. Choosing an identification , this is given by mutually commuting invertible matrices . In order to extend we need to construct a logarithmic connection on the trivial bundle which has the same monodromy when restricted to (since this will allow the map to be extended to a well-defined isomorphism of flat bundles ).
To this end we make use of the theorem on matrix logarithms which states that there are unique matrices such that
- the eigenvalues of lie in the image of .
Note that the matrices mutually commute since the do.
Now define the connection on as follows
which has logarithmic poles along . The residue along the component is given by
which by hypothesis has its eigenvalues in the image of . A fundamental solution to (i.e. a matrix-valued function whose columns are flat sections; this can be used to define the parallel transport operator) is given by
and so the monodromy around loop is given by
Hence is the desired extension.
Let be another connection satisfying the properties of the theorem. We want to show that there is a unique isomorphism between this flat vector bundle and the extension constructed above. Since both and extend from to they are already identified over :
Hence an isomorphism between these flat bundles must be an isomorphism extending to , and this is clearly unique if it exists. Therefore we only need to show that we can holomorphically extend (and ) to . We will only show that extends since the situation with is identical.
We note a further simplification: if we can show that extends holomorphically to then by Hartogs’ theorem it automatically extends to . Hence it suffices to assume that has a single component: .
Now in order to show that extends holomorphically we will derive a differential equation that it must satisfy. Recall what it means for to be a morphism of flat bundles: . Writing and this condition can be expressed as follows:
In a neighbourhood of a point on the divisor we can write
for holomorphic matrix valued functions . Plugging this, and the expression for , into equation 1 and taking the coefficient of we get
Taking () norms of both sides of this equation, and noting that since and are holomorphic they are bounded in some neighbourhood of every point on the divisor, we get
for some constant . For points away from the divisor we can write and we have
where is the entry of the matrix . By the fundamental theorem of calculus we have for
and taking absolute values and using the above inequality gives
Finally, if we add up the contributions from all components of we end up with the following inequality for the norm of :
for positive constants . We can now apply the integral form of Grönwall’s inequality to deduce that there exists a positive constant and a positive integer such that for small enough we have
By a version of the Riemann removable singularities theorem for many variables this implies that is meromorphic at . Hence we can write a series expansion for as follows:
where are holomorphic matrix-valued functions on and is non-zero (i.e. we take to be the smallest integer such that ). Plugging this expression for back into equation 2 and taking the coefficient of yields the following equation
which, upon rearranging and observing that and , becomes
If was invertible then and would be conjugate and hence have the same spectrum. At present we only know that and so we are only guaranteed that they have an eigenvalue in common. But both residues have their eigenvalues in the image of , and so this implies that . This shows that extends holomorphically.