Deligne’s construction for extending connections.

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 $X$ be a smooth complex manifold, and let $Y \subseteq X$ be a divisor (i.e. a codimension 1 subvariety). Then given a branch of the logarithm $\tau$ (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 $Y$ to a holomorphic vector bundle on $X$ with a flat logarithmic connection having poles along $Y$ :

Theorem. Let $(F,\nabla)$ be a holomorphic vector bundle with flat connection on $X \setminus Y$ . There exists a unique extension $(G,\tilde{\nabla})$ to all of $X$ such that $\tilde{\nabla}$ has logarithmic singularities along $Y \subseteq X$ and the eigenvalues of the residue of $\tilde{\nabla}$ lie in the image of $\tau$ .

1. Preliminaries on connections.

Throughout let ${X}$ be a complex manifold, and ${F \rightarrow X}$ a holomorphic vector bundle.

Definition.

A connection on ${F}$ is a ${\mathbb{C}_{X}}$ -linear map $\displaystyle \nabla : F \rightarrow F \otimes \Omega_{X}^{1},$
satisfying the Liebniz rule: for every function ${\phi \in \mathcal{O}_{X}}$ and section ${f \in F}$ $\displaystyle \nabla(\phi f) = d \phi \otimes f + \phi \nabla f.$
Given a vector field ${V \in \mathcal{T}_{X}}$ , we can hook it into the 1-form part of the connection to get a ${\mathbb{C}}$ -linear map ${\nabla_{V} : F \rightarrow F}$ . The connection is flat if for all vector fields ${V,U \in \mathcal{T}_{X}}$ : $\displaystyle [ \nabla_{V}, \nabla_{U}] = \nabla_{[V,U]}.$

Remark. By hooking vector fields into the connection, we see that a connection defines an action of the tangent sheaf of ${X}$ on the vector bundle : $\nabla : \mathcal{T}_{X} \to End_{\mathbb{C}}(F)$ . The condition that ${\nabla}$ 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 ${\mathcal{T}_{X}}$ .

Locally, we can pick a frame for ${F}$ , allowing us to write a connection as

$\displaystyle \nabla = d + \omega,$

for a matrix valued 1-form ${\omega}$ . More precisely let ${f_{1}, ..., f_{m}}$ be a local frame. Then ${\nabla f_{j} = \sum_{i} \omega_{ij} f_{i}}$ , for 1-forms ${\omega_{ij}}$ . Then writing a general section in terms of the frame as a column vector ${\phi}$ we have

$\displaystyle \nabla \phi = d\phi + \omega \phi,$

where ${\omega = (\omega_{ij})_{ij}}$ .

It turns out that, despite being defined in terms of a complex structure on ${X}$ and a holomorphic structure on ${F}$ , 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 ${X}$ :

$\displaystyle \text{Rep}(\mathcal{T}_{X}) \longleftrightarrow \text{Rep}(\pi_{1}(X)).$

The equivalence is given by sending a flat connection to its monodromy representation.

Sketch of Proof. Given a bundle with flat connection ${(F, \nabla)}$ , we use the flat sections (i.e. ${f \in F}$ such that ${\nabla f = 0}$ ) to identify nearby fibers of ${F}$ . This allows us to define a parallel transport operator ${P}$ , which associates a linear isomorphism to each homotopy class of paths:

$\displaystyle [\gamma] \mapsto P_{\gamma} : F_{\gamma(0)} \xrightarrow{\sim} F_{\gamma(1)}.$

Picking a base-point ${x \in X}$ and applying the parallel transport operator to the loops based at ${x}$ then gives the monodromy representation:

$\displaystyle P|_{\pi_{1}(X,x)} : \pi_{1}(X,x) \rightarrow Gl(F_{x}).$

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 ${X}$

$\displaystyle \Pi_{1}(X) := \{ \text{homotopy classes of paths in } X\}$

As it turns out, this representation is nothing but the parallel transport operator ${P}$ defined above. Finally, since ${\Pi_{1}(X)}$ is transitive, it is Morita equivalent to the fundamental group of ${X}$ (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

$\displaystyle \text{Rep}(\mathcal{T}_{X}) \underset{\text{Lie}}{\longleftrightarrow} \text{Rep}(\Pi_{1}(X)) \underset{\text{Morita}}{\longleftrightarrow} \text{Rep}(\pi_{1}(X)).$

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 ${Y \subset X}$ 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 ${x_{1}, ..., x_{n}}$ such that ${Y = \{x_{1}...x_{p} = 0 \}}$ ). Let ${Y_{i}}$ denote the irreducible components of ${Y}$ .

Definition.

The sheaf of logarithmic tangent vectors ${\mathcal{T}_{X}(-\log Y)}$ is the subsheaf of ${\mathcal{T}_{X}}$ consisting of vector fields on ${X}$ which are tangent to ${Y}$ along ${Y}$ . This sheaf is locally free and hence defines a vector bundle ${T_{X}(-\log Y)}$ . 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 $\displaystyle \Omega_{X}^{1}(\log Y) := (\mathcal{T}_{X}(- \log Y))^{\vee}.$

Given local coordinates ${x_{1}, ..., x_{n}}$ such that ${Y = \{ x_{1} = 0 \}}$ we have

$\displaystyle \mathcal{T}_{X}(-\log Y) = \langle x_{1} \partial_{x_{1}}, \partial_{x_{2}}, ... \rangle, \ \Omega_{X}^{1}(\log Y) = \langle \frac{dx_{1}}{x_{1}}, dx_{2}, ... \rangle.$

Proposition. Suppose ${Y}$ is smooth. The sheaf of 1-forms with logarithmic poles lies in the following short exact sequence

$\displaystyle 0 \rightarrow \Omega_{X}^{1} \rightarrow \Omega_{X}^{1}(\log Y) \overset{\text{Res}}{\rightarrow} \mathcal{O}_{Y} \rightarrow 0,$

where ${\text{Res} : \Omega_{X}^{1}(\log Y) \rightarrow \mathcal{O}_{Y}}$ is the residue map, given locally by

$\displaystyle \text{Res}(a_{1} \frac{dx_{1}}{x_{1}} + a_{2} dx_{2} + ... ) = a_{1}|_{Y}.$

Remark. In the case that ${Y}$ is a divisor with normal crossings we can define a residue map ${\text{Res}_{i}}$ for each component ${Y_{i}}$ .

Definition.

A connection on ${F \rightarrow X}$ with logarithmic poles along ${Y}$ is a ${\mathbb{C}_{X}}$ -linear map $\displaystyle \nabla : F \rightarrow F\otimes \Omega_{X}^{1}(\log Y)$ satisfying the Liebniz rule. The notion of flatness still makes sense in this context. Note that a flat connection with logarithmic poles along ${Y}$ defines a representation of the logarithmic tangent sheaf.
The residue of a logarithmic connection ${\nabla}$ along ${Y}$ is the endomorphism ${\text{Res}{\nabla} \in End_{\mathcal{O}_{Y}}(F|_{Y})}$ defined by composing the connection with the residue on logarithmic forms: $\displaystyle F \overset{\nabla}{\rightarrow} F \otimes \Omega_{X}^{1}(\log Y) \overset{\text{Res}}{\rightarrow} F \otimes \mathcal{O}_{Y}.$

Remark. When ${Y}$ is a normal crossing divisor we define the residue for each component ${Y_{i}}$ .

Given coordinates for ${X}$ and a choice of frame for ${F}$ as above we can write a logarithmic connection as

$\displaystyle \nabla = d + \frac{M_{1}}{x_{1}}dx_{1} + \sum_{i = 2}^{n} M_{i}dx_{i},$

for holomorphic matrix-valued functions ${M_{i}}$ . The residue for this connection is given by ${M_{1}|_{Y}}$ .

3. Deligne’s construction for extending connections

Consider the exponential sequence

$\displaystyle 0 \rightarrow \mathbb{Z} \rightarrow \mathbb{C} \overset{e^{2 \pi i }}{\rightarrow} \mathbb{C}^{\times} \rightarrow O,$

and choose a set-theoretical splitting ${\tau : \mathbb{C}^{\times} \rightarrow \mathbb{C}}$ . Then we have the following theorem which is due to Deligne:

Theorem. Let ${F \rightarrow (X \setminus Y)}$ be a holomorphic vector bundle, and ${\nabla}$ a holomorphic flat connection on ${F}$ . Then there exists a holomorphic vector bundle with flat logarithmic connection ${(G,\tilde{\nabla})}$ extending ${(F, \nabla)}$ to ${X}$ , unique up to unique isomorphism, such that

${\tilde{\nabla}}$ has logarithmic poles along ${Y}$ ,
the eigenvalues of the residues of ${\tilde{\nabla}}$ lie in the image of the splitting ${\tau}$ .

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 ${X}$ is a polydisc:

$\displaystyle X = \{ (x_{1},...,x_{n}) \ | \ |x_{i}| < r_{i} \} = D_{r_{1}} \times ... \times D_{r_{n}},$

and that ${Y = \{ x_{1}...x_{p} = 0 \}}$ , with ith component ${Y_{i} = \{ x_{i} = 0 \}}$ . The proof now proceeds in two parts: existence and uniqueness.

3.1. Existence

${(F, \nabla )}$ is a holomorphic flat connection on ${X \setminus Y}$ and so by the Riemann-Hilbert correspondence it is determined by it’s monodromy representation. First note that since

$\displaystyle X \setminus Y = (D_{r_{1}} \setminus \{ 0 \}) \times ... \times (D_{r_{p}} \setminus \{ 0 \})\times D_{r_{p+1}} \times ... \times D_{r_{n}},$

it follows that ${\pi_{1}(X \setminus Y) \cong \mathbb{Z}^{p}}$ . Let us make some choices in order to exhibit this isomorphism: choose a point ${a \in X \setminus Y}$ . Then the fundamental group is generated by ${p}$ loops, and we take the ith loop to be the loop ${\gamma_{i}}$ which starts at ${a}$ and goes around ${Y_{i}}$ counterclockwise in the plane ${\{x_{1} = a_{1}, ..., x_{i-1} = a_{i-1}, x_{i+1} = a_{i+1}, ..., x_{n} = a_{n}\}}$ . The monodromy representation is given by taking the parallel transport around each of these loops. Choosing an identification ${\mathbb{C}^{m} \overset{\sim}{\rightarrow} F_{a}}$ , this is given by ${p}$ mutually commuting invertible matrices ${C_{1}, ..., C_{p} \in Gl(\mathbb{C}^m)}$ . In order to extend ${(F, \nabla)}$ we need to construct a logarithmic connection on the trivial bundle ${G = \mathbb{C}^{m} \times X}$ which has the same monodromy when restricted to ${X \setminus Y}$ (since this will allow the map ${\mathbb{C}^{m} \overset{\sim}{\rightarrow} F_{a}}$ to be extended to a well-defined isomorphism of flat bundles ${G|_{X \setminus Y} \overset{\sim}{\rightarrow} F}$ ).

To this end we make use of the theorem on matrix logarithms which states that there are unique matrices ${\Gamma_{1}, ..., \Gamma_{p} \in End(\mathbb{C}^{m})}$ such that

${\exp(-2 \pi i \Gamma_{i}) = C_{i}}$ ,
the eigenvalues of ${\Gamma_{i}}$ lie in the image of ${\tau}$ .

Note that the matrices ${\Gamma_{i}}$ mutually commute since the ${C_{i}}$ do.

Now define the connection on ${G}$ as follows

$\displaystyle \tilde{\nabla} = d + \sum_{i = 1}^{p} \Gamma_{i} \frac{dx_{i}}{x_{i}},$

which has logarithmic poles along ${Y}$ . The residue along the component ${Y_{i}}$ is given by

$\displaystyle \text{Res}_{i}\tilde{\nabla} = \Gamma_{i},$

which by hypothesis has its eigenvalues in the image of ${\tau}$ . A fundamental solution to ${\tilde{\nabla}}$ (i.e. a matrix-valued function whose columns are flat sections; this can be used to define the parallel transport operator) is given by

$\displaystyle \Phi(x) = \exp(-\sum_{i = 1}^{p} \Gamma_{i} \log(x_{i})) = \Pi_{i = 1}^{p} x_{i}^{-\Gamma_{i}},$

and so the monodromy around loop ${\gamma_{i}}$ is given by

$\displaystyle \Phi((a_{1},..., a_{i}e^{2 \pi i}, ...)) \Phi(a)^{-1} = \exp(-2 \pi i \Gamma_{i}) = C_{i}.$

Hence ${(G, \tilde{\nabla})}$ is the desired extension.

3.2. Uniqueness

Let ${(G', \tilde{\nabla}')}$ 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 ${(G, \tilde{\nabla})}$ constructed above. Since both ${(G, \tilde{\nabla})}$ and ${(G', \tilde{\nabla}')}$ extend ${(F,\nabla)}$ from ${X \setminus Y}$ to ${X}$ they are already identified over ${X \setminus Y}$ :

$\displaystyle S : (G, \tilde{\nabla})|_{X \setminus Y} \overset{\sim}{\rightarrow} (G', \tilde{\nabla}')|_{X \setminus Y}.$

Hence an isomorphism between these flat bundles must be an isomorphism extending ${S}$ to ${X}$ , and this is clearly unique if it exists. Therefore we only need to show that we can holomorphically extend ${S}$ (and ${S^{-1}}$ ) to ${X}$ . We will only show that ${S}$ extends since the situation with ${S^{-1}}$ is identical.

We note a further simplification: if we can show that ${S}$ extends holomorphically to ${Y_{i} \setminus \cup_{j \neq i} Y_{j}}$ then by Hartogs’ theorem it automatically extends to ${Y_{i}}$ . Hence it suffices to assume that ${Y}$ has a single component: ${Y = \{ x_{1} = 0 \}}$ .

Now in order to show that ${S}$ extends holomorphically we will derive a differential equation that it must satisfy. Recall what it means for ${S}$ to be a morphism of flat bundles: ${S \tilde{\nabla} = \tilde{\nabla}' S}$ . Writing ${\tilde{\nabla} = d + \omega}$ and ${\tilde{\nabla}' = d + \omega'}$ this condition can be expressed as follows:

$\displaystyle dS = S \omega - \omega' S. \ \ \ \ \ (1)$

In a neighbourhood of a point on the divisor we can write

$\displaystyle \omega' = M_{1}' \frac{dx_{1}}{x_{1}} + \sum_{i = 2}^{n} M_{i}'dx_{i}$

for holomorphic matrix valued functions ${M_{i}'}$ . Plugging this, and the expression for ${\omega}$ , into equation 1 and taking the coefficient of ${dx_{1}}$ we get

$\displaystyle x_{1} \frac{\partial S}{\partial x_{1}} = S \Gamma_{1} - M_{1}' S. \ \ \ \ \ (2)$

Taking ( ${l^2}$ ) norms of both sides of this equation, and noting that since ${\Gamma_{1}}$ and ${M_{1}'}$ are holomorphic they are bounded in some neighbourhood of every point on the divisor, we get

$\displaystyle |x_{1}| ||\frac{\partial S}{\partial x_{1}}|| \leq ||S \Gamma_{1} || + ||M_{1}' S|| \leq C ||S||,$

for some constant ${C}$ . For points away from the divisor we can write ${x_{1} = r e^{i \theta}}$ and we have

$\displaystyle |\frac{\partial S_{ij}}{\partial r}| = |\frac{\partial S_{ij}}{\partial x_{1}}| \leq ||\frac{\partial S}{\partial x_{1}}|| \leq \frac{C}{r} ||S||,$

where $\displaystyle S_{ij}$ is the $\displaystyle ij$ entry of the matrix $\displaystyle S$ . By the fundamental theorem of calculus we have for ${0 < r < r_{0}}$

$\displaystyle S_{ij}(r e^{i \theta}) = S_{ij}(r_{0} e^{i \theta}) + \int_{r_{0}}^{r} \frac{\partial S_{ij}}{\partial s} ds,$

and taking absolute values and using the above inequality gives

$\displaystyle |S_{ij}(r e^{i \theta})| \leq |S_{ij}(r_{0} e^{i \theta})| + \int_{r_{0}}^{r} \frac{C}{s} ||S(s e^{i \theta})|| ds.$

Finally, if we add up the contributions from all components of ${S}$ we end up with the following inequality for the norm of ${S}$ :

$\displaystyle ||S(r e^{i \theta}) || \leq K_{1} + \int_{r}^{r_{0}} \frac{K_{2}}{s} ||S(s e^{i \theta})|| ds,$

for positive constants ${K_{1}, K_{2}}$ . We can now apply the integral form of Grönwall’s inequality to deduce that there exists a positive constant ${K}$ and a positive integer ${N}$ such that for ${|x_{1}|}$ small enough we have

$\displaystyle ||S(x)|| \leq K |x_{1}|^{-N}.$

By a version of the Riemann removable singularities theorem for many variables this implies that ${S}$ is meromorphic at ${Y}$ . Hence we can write a series expansion for ${S}$ as follows:

$\displaystyle S(x) = \sum_{k = q}^{\infty} x_{1}^{k} S_{k}(x_{2},...,x_{n}),$

where ${S_{k}}$ are holomorphic matrix-valued functions on ${Y}$ and ${S_{q}}$ is non-zero (i.e. we take ${q}$ to be the smallest integer such that ${S_{q} \neq 0}$ ). Plugging this expression for ${S}$ back into equation 2 and taking the coefficient of ${x_{1}^{q}}$ yields the following equation

$\displaystyle q S_{q} = S_{q} \Gamma_{1} - M_{1}'|_{Y} S_{q},$

which, upon rearranging and observing that ${\Gamma_{1} = \text{Res} \tilde{\nabla}}$ and ${M_{1}'|_{Y} = \text{Res} \tilde{\nabla}'}$ , becomes

$\displaystyle (\text{Res} \tilde{\nabla}' + q I) S_{q} = S_{q} \text{Res} \tilde{\nabla}.$

If ${S_{q}}$ was invertible then ${\text{Res} \tilde{\nabla}}$ and ${(\text{Res} \tilde{\nabla}' + q I)}$ would be conjugate and hence have the same spectrum. At present we only know that ${S_{q} \neq 0}$ and so we are only guaranteed that they have an eigenvalue in common. But both residues have their eigenvalues in the image of ${\tau}$ , and so this implies that ${q = 0}$ . This shows that ${S}$ extends holomorphically. ${\Box}$

Deligne’s construction for extending connections.

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